Explanation:
Okay, so I need to find the area of the quadrilateral formed by the four points of tangency between the circle x² + y² = 2 and the parabola y² = 8x. The problem mentions there are two common tangents, which form four points of tangency. Wait, hold on, two common tangents? But tangents usually touch at one point each. If there are two common tangents, each tangent would have one point of tangency, so that would be two points. But the problem says four points of tangency. Maybe I need to clarify that.

Wait, maybe the circle and the parabola have two common tangents, but each tangent touches both the circle and the parabola? So each tangent line is tangent to both the circle and the parabola. Therefore, each tangent line would have one point of tangency on the circle and one on the parabola. Therefore, two common tangents would result in two points on the circle and two points on the parabola? But the problem says four points of tangency. Maybe there are two common tangents, each tangent to both the circle and the parabola, so each tangent has two points of tangency? Wait, no. A tangent line to a curve touches it at one point. So if there are two common tangents, each tangent line touches the circle at one point and the parabola at another point. Therefore, there are two points on the circle and two points on the parabola. But the problem says four points of tangency. Maybe the problem is referring to the four points where the common tangents touch the circle and the parabola? So two tangents, each touching the circle once and the parabola once, so total four points. Then the quadrilateral is formed by these four points. So the four points are two on the circle and two on the parabola? Or maybe the tangents are such that each tangent touches both curves at the same point? But that seems unlikely unless they intersect at a point where they are tangent to both. But the circle and parabola might intersect? Let me check.

First, let me check if the circle and parabola intersect. The circle is x² + y² = 2, and the parabola is y² = 8x. Substitute y² from the parabola into the circle equation: x² + 8x = 2 => x² + 8x - 2 = 0. The discriminant is 64 + 8 = 72, so there are two real solutions. Therefore, the circle and parabola intersect at two points. Therefore, they are not tangent to each other. Therefore, the common tangents are lines that touch both the circle and the parabola without intersecting them. Therefore, each common tangent touches the circle at one point and the parabola at another point. Therefore, there are two common tangents, each contributing one point on the circle and one on the parabola, so four points total. Then the quadrilateral is formed by connecting these four points. Wait, but how? If there are two points on the circle and two on the parabola, how do they form a quadrilateral? Maybe the four points are two on the circle and two on the parabola, but arranged such that they form a quadrilateral. But maybe the problem is referring to the points where the common tangents touch the curves. So each tangent line touches the circle and the parabola, so each tangent line has two points of tangency. Therefore, two tangents would have four points of tangency. Therefore, the quadrilateral is formed by connecting these four points. So the four points are two on the circle and two on the parabola. But how do they form a quadrilateral? Maybe the two tangents are symmetric with respect to the x-axis? Let me think.

Alternatively, maybe there are four common tangents? Wait, the problem says two common tangents. Wait, maybe the Asymptote code gives some clue. The Asymptote code draws the upper and lower parabola branches, the circle, and some lines. The points A, B, C, D are given as (-1,1), (2,4), (-1,-1), (2,-4). Then it draws the graph of the upper and lower parabola, the circle, and some lines connecting A to B and C to D, and also A to C and B to D. The points A, B, C, D are probably the points of tangency. So maybe the quadrilateral is A, B, C, D? Let me check coordinates. If A is (-1,1), then plugging into the circle equation: (-1)^2 + (1)^2 = 1 + 1 = 2, which satisfies the circle equation. Similarly, point C is (-1,-1), which also satisfies the circle equation. Then points B (2,4): 2² + 4² = 4 + 16 = 20 ≠ 2. Wait, that's not on the circle. Wait, but the problem says the quadrilateral is formed by the four points of tangency. But according to the Asymptote code, points A and C are on the circle, but points B and D are not. So maybe the Asymptote code is illustrative but not exact? Or maybe the coordinates given in the Asymptote code are not the actual points of tangency? Because if points B and D are on the parabola, let's check: for point B (2,4), y² = 16, and 8x = 16, so yes, that's on the parabola. Similarly, D (2,-4) is also on the parabola. So points A and C are on the circle, points B and D are on the parabola. Then the quadrilateral is formed by connecting A, B, C, D? But how? If you connect A to B to C to D to A? Let me visualize. A is (-1,1), B is (2,4), C is (-1,-1), D is (2,-4). So connecting A(-1,1) to B(2,4), then to C(-1,-1), then to D(2,-4), then back to A? That seems like a quadrilateral. But maybe the actual points of tangency are different? Wait, but the Asymptote code is just an illustration. The problem says "the quadrilateral formed by the four points of tangency," so maybe the answer is the area of quadrilateral ABCD as given? But the coordinates given in the Asymptote code are probably not the actual points of tangency. Because if you check the tangent lines from A to B, but A is on the circle and B is on the parabola. Wait, but the problem says the common tangents. So each tangent line is tangent to both the circle and the parabola. Therefore, each tangent line touches the circle at one point and the parabola at another. Therefore, there are two such tangent lines, each contributing two points of tangency. Therefore, four points total. So the quadrilateral is formed by these four points. Therefore, the Asymptote code's points A, B, C, D are probably the actual points of tangency. But in that case, the coordinates given in the Asymptote code are (-1,1), (2,4), (-1,-1), (2,-4). Let me check if these points are points of tangency. For example, take point A (-1,1). Is there a tangent line to both the circle and the parabola that touches the circle at A and the parabola at B (2,4)? Let me check if the line AB is tangent to both curves.

First, find the equation of the line AB. Points A(-1,1) and B(2,4). The slope is (4 - 1)/(2 - (-1)) = 3/3 = 1. So the equation is y - 1 = 1*(x + 1), so y = x + 2. Let me check if this line is tangent to the circle x² + y² = 2. Substitute y = x + 2 into the circle equation:

x² + (x + 2)^2 = 2 => x² + x² + 4x + 4 = 2 => 2x² + 4x + 2 = 0 => x² + 2x + 1 = 0 => (x + 1)^2 = 0. So x = -1, which is the point A(-1,1). Therefore, the line AB is tangent to the circle at A. Now check if it's tangent to the parabola y² = 8x. Substitute y = x + 2 into the parabola equation:

(x + 2)^2 = 8x => x² + 4x + 4 = 8x => x² - 4x + 4 = 0 => (x - 2)^2 = 0 => x = 2. Therefore, the line AB touches the parabola at x = 2, y = 4, which is point B. Therefore, the line AB is indeed a common tangent to both the circle and the parabola, touching the circle at A and the parabola at B. Similarly, the line CD connects C(-1,-1) and D(2,-4). Let me check that. The slope is (-4 - (-1))/(2 - (-1)) = (-3)/3 = -1. Equation: y - (-1) = -1(x - (-1)) => y + 1 = -x -1 => y = -x - 2. Check if this is tangent to the circle. Substitute y = -x - 2 into x² + y² = 2:

x² + (-x - 2)^2 = 2 => x² + x² + 4x + 4 = 2 => 2x² + 4x + 2 = 0 => same as before, x = -1. Therefore, tangent at point C(-1,-1). Check if tangent to parabola y² = 8x. Substitute y = -x - 2:

(-x - 2)^2 = 8x => x² + 4x + 4 = 8x => x² - 4x + 4 = 0 => (x - 2)^2 = 0 => x = 2. Therefore, tangent at point D(2,-4). Therefore, the line CD is also a common tangent. Therefore, the four points of tangency are A(-1,1), B(2,4), C(-1,-1), D(2,-4). Therefore, the quadrilateral is formed by these four points. Therefore, the area of quadrilateral ABCD is the area we need to find. However, the Asymptote code also draws lines connecting A to B, C to D, and A to C, B to D. So the quadrilateral is probably a trapezoid or some quadrilateral. Let me plot these points mentally. Points A(-1,1), B(2,4), C(-1,-1), D(2,-4). So plotting these, A is on the left upper, B is on the right upper, C is on the left lower, D is on the right lower. Connecting A to B to C to D to A. This seems like a trapezoid? Let me check if sides AB and CD are parallel. The slope of AB is 1, slope of CD is -1. Not parallel. The slope of BC: from B(2,4) to C(-1,-1): slope is (-1 - 4)/(-1 - 2) = (-5)/(-3) = 5/3. Slope of DA: from D(2,-4) to A(-1,1): slope is (1 - (-4))/(-1 - 2) = 5/(-3) = -5/3. Not parallel. So maybe it's a kite? Or some other quadrilateral. Alternatively, maybe it's a parallelogram? Wait, no. Alternatively, maybe it's symmetric with respect to the x-axis. Let me check coordinates. Points A(-1,1) and C(-1,-1) are symmetric over the x-axis. Similarly, points B(2,4) and D(2,-4) are symmetric over the x-axis. Therefore, the quadrilateral is symmetric over the x-axis. Therefore, maybe we can compute the area by computing the area of the upper half and doubling it. Let me think. Alternatively, since it's symmetric, maybe we can use coordinates to compute the area.

Alternatively, since we have coordinates of all four points, we can use the shoelace formula. Let me recall the shoelace formula. For a polygon with vertices (x1,y1), (x2,y2), ..., (xn,yn), the area is |1/2 * sum_{i=1 to n} (xi*yi+1 - xi+1*yi)|, where xn+1=x1, yn+1=y1.

So let's apply that. Let me list the coordinates in order. Let's take A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A(-1,1).

Compute the terms:

x1*y2 - x2*y1: (-1)*4 - 2*1 = -4 - 2 = -6

x2*y3 - x3*y2: 2*(-1) - (-1)*4 = -2 + 4 = 2

x3*y4 - x4*y3: (-1)*(-4) - 2*(-1) = 4 + 2 = 6

x4*y1 - x1*y4: 2*1 - (-1)*(-4) = 2 - 4 = -2

Sum these up: -6 + 2 + 6 - 2 = 0? Wait, that can't be. Wait, maybe I did the shoelace formula wrong. Let me check again.

Wait, the shoelace formula is sum over (xi * yi+1 - xi+1 * yi). So let me write down all terms step by step.

Order of points: A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A(-1,1).

Compute each term:

First term: x_A * y_B - x_B * y_A = (-1)(4) - (2)(1) = -4 - 2 = -6

Second term: x_B * y_C - x_C * y_B = (2)(-1) - (-1)(4) = -2 + 4 = 2

Third term: x_C * y_D - x_D * y_C = (-1)(-4) - (2)(-1) = 4 + 2 = 6

Fourth term: x_D * y_A - x_A * y_D = (2)(1) - (-1)(-4) = 2 - 4 = -2

Sum of all terms: -6 + 2 + 6 - 2 = 0. Wait, that gives zero? That can't be. That would mean the area is zero, which is impossible. So I must have messed up the order of the points. Maybe the points are not ordered correctly for the shoelace formula. Let me check the order. If I connect A(-1,1) to B(2,4) to C(-1,-1) to D(2,-4) to A(-1,1), is that a correct order? Let me visualize again. If you go from A(-1,1) to B(2,4), then to C(-1,-1), that would cross over? Maybe the order is not correct. Maybe I need to order the points in a cyclic order around the quadrilateral. Let me think. If the quadrilateral is symmetric over the x-axis, maybe the correct order is A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A. Let me try that.

Order: A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A(-1,1).

Compute terms:

First term: x_A * y_B - x_B * y_A = (-1)(4) - 2(1) = -4 - 2 = -6

Second term: x_B * y_D - x_D * y_B = 2*(-4) - 2*4 = -8 - 8 = -16

Third term: x_D * y_C - x_C * y_D = 2*(-1) - (-1)*(-4) = -2 - 4 = -6

Fourth term: x_C * y_A - x_A * y_C = (-1)(1) - (-1)*(-1) = -1 - 1 = -2

Sum: -6 -16 -6 -2 = -30. Take absolute value and divide by 2: | -30 | / 2 = 15. So area is 15? Wait, but let me check again. Maybe I made a mistake in the order.

Alternatively, maybe the correct order is A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A. But that gave zero. Maybe I need to arrange the points in a different order. Let me try another order. Let me think of the quadrilateral as two triangles. For example, A, B, D, C. Wait, maybe the quadrilateral is actually a parallelogram? Wait, if we connect A(-1,1) to B(2,4) to D(2,-4) to C(-1,-1) to A. Let me check if AB is parallel to CD? Wait, AB has slope 1, CD is from C(-1,-1) to D(2,-4): slope is (-4 - (-1))/(2 - (-1)) = (-3)/3 = -1. Not parallel. AD is from A(-1,1) to D(2,-4): slope is (-4 - 1)/(2 - (-1)) = -5/3. BC is from B(2,4) to C(-1,-1): slope is (-1 - 4)/(-1 - 2) = -5/-3 = 5/3. So AD and BC have slopes -5/3 and 5/3, which are negative reciprocals? No, they are negative of each other. So if AB is slope 1, BC is 5/3, CD is -1, DA is -5/3. Not sure. Maybe it's a trapezoid? Wait, if AB and CD are not parallel, and AD and BC are not parallel. So maybe it's a general quadrilateral. Let me try another order. Maybe A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A. Let me recalculate the shoelace formula with this order.

Coordinates:

A(-1,1), B(2,4), D(2,-4), C(-1,-1), A(-1,1)

Compute terms:

First term: x_A * y_B - x_B * y_A = (-1)(4) - 2(1) = -4 - 2 = -6

Second term: x_B * y_D - x_D * y_B = 2*(-4) - 2*4 = -8 - 8 = -16

Third term: x_D * y_C - x_C * y_D = 2*(-1) - (-1)*(-4) = -2 - 4 = -6

Fourth term: x_C * y_A - x_A * y_C = (-1)(1) - (-1)*(-1) = -1 - 1 = -2

Total sum: -6 -16 -6 -2 = -30. Absolute value divided by 2: 15. So area is 15? But let me check with another order. Suppose I take A(-1,1), C(-1,-1), D(2,-4), B(2,4), back to A. Let me compute:

First term: x_A * y_C - x_C * y_A = (-1)(-1) - (-1)(1) = 1 + 1 = 2

Second term: x_C * y_D - x_D * y_C = (-1)(-4) - 2*(-1) = 4 + 2 = 6

Third term: x_D * y_B - x_B * y_D = 2*4 - 2*(-4) = 8 + 8 = 16

Fourth term: x_B * y_A - x_A * y_B = 2*1 - (-1)*4 = 2 + 4 = 6

Total sum: 2 + 6 + 16 + 6 = 30. Absolute value divided by 2: 15. Same result. So area is 15? But let me check with coordinates. Alternatively, since the quadrilateral is symmetric over the x-axis, maybe we can compute the area of the upper half and double it. The upper half would be the triangle ABD? Wait, no. Wait, points A(-1,1), B(2,4), and the reflection of D is B? Wait, no. Wait, the upper half would be the area between A(-1,1), B(2,4), and the x-axis? Maybe not. Alternatively, since the quadrilateral is symmetric over the x-axis, we can compute the area of the upper half (the part above the x-axis) and double it. The upper half would be the triangle formed by A(-1,1), B(2,4), and the projection on the x-axis? Wait, maybe not. Alternatively, think of the quadrilateral as two congruent trapezoids? Wait, maybe not. Alternatively, since we have coordinates, maybe using coordinates is the safest way. But according to shoelace formula, the area is 15. However, let me check with another method. For example, coordinates of A(-1,1), B(2,4), C(-1,-1), D(2,-4). Let me compute vectors or use coordinates.

Alternatively, since the quadrilateral is symmetric over the x-axis, we can compute the area of the upper half and double it. The upper half consists of points A(-1,1), B(2,4), and the reflection of C and D over the x-axis. Wait, but since it's symmetric, the upper half would be a triangle or a trapezoid. Wait, if we take the upper half, it's the area between A(-1,1), B(2,4), and the x-axis? Wait, no. The quadrilateral is formed by A, B, C, D. If we split it along the x-axis, we have two congruent parts. The upper part is the triangle ABD? Wait, no. Wait, if you imagine the quadrilateral, it's like a bowtie? Wait, no. Wait, if you connect A(-1,1) to B(2,4) to D(2,-4) to C(-1,-1) to A, then it's a non-intersecting quadrilateral? Wait, connecting A to B to D to C to A. Let me check if the sides cross. From A(-1,1) to B(2,4), then to D(2,-4): that's a line from B(2,4) down to D(2,-4), which is vertical line x=2. Then from D(2,-4) to C(-1,-1), then back to A(-1,1). So this seems like a quadrilateral that is a trapezoid? Wait, but with one vertical side? Wait, BD is vertical. Then sides AB and DC? Wait, AB is from (-1,1) to (2,4), DC is from (2,-4) to (-1,-1). Are AB and DC parallel? Let me check their slopes. AB slope is 1, DC slope is (-1 - (-4))/(-1 - 2) = (3)/(-3) = -1. Not parallel. So not a trapezoid. So maybe it's a kite? Let me check. A kite has two pairs of adjacent sides equal. Let me compute lengths.

AB: distance between A(-1,1) and B(2,4): sqrt[(2 - (-1))² + (4 - 1)²] = sqrt[9 + 9] = sqrt(18) = 3√2.

AD: distance between A(-1,1) and D(2,-4): sqrt[(2 - (-1))² + (-4 - 1)²] = sqrt[9 + 25] = sqrt(34).

BC: distance between B(2,4) and C(-1,-1): sqrt[(-1 - 2)^2 + (-1 - 4)^2] = sqrt[9 + 25] = sqrt(34).

CD: distance between C(-1,-1) and D(2,-4): sqrt[(2 - (-1))² + (-4 - (-1))²] = sqrt[9 + 9] = sqrt(18) = 3√2.

So sides AB and CD are equal (3√2), and sides AD and BC are equal (sqrt(34)). Therefore, it is a kite. Therefore, the area of a kite is (d1*d2)/2, where d1 and d2 are the lengths of the diagonals. Let me find the diagonals. The diagonals are AC and BD.

Diagonal AC: from A(-1,1) to C(-1,-1). This is vertical line x=-1, from y=1 to y=-1. Length is 2 units.

Diagonal BD: from B(2,4) to D(2,-4). This is vertical line x=2, from y=4 to y=-4. Length is 8 units.

Wait, so diagonals are AC (length 2) and BD (length 8). Therefore, area is (2 * 8)/2 = 8. Wait, but earlier shoelace formula gave 15? There's a contradiction here. Which one is correct?

Wait, if it's a kite with diagonals 2 and 8, area is 8. But shoelace formula gave 15. There must be a mistake in one of the methods. Let me check the diagonals again. Wait, in a kite, the diagonals are perpendicular? Wait, in a kite, one diagonal is the axis of symmetry. In this case, since the kite is symmetric over the x-axis, the diagonals should be AC and BD. Wait, AC is vertical line x=-1, BD is vertical line x=2. Wait, but they are both vertical, so they are parallel? That can't be. Wait, if both diagonals are vertical, then they are parallel, which would mean the kite is degenerate? But that can't be. Wait, something is wrong here. Wait, if the diagonals are AC and BD, but AC is from (-1,1) to (-1,-1), BD is from (2,4) to (2,-4). These are two vertical lines, parallel, so they don't intersect. Therefore, this is not a kite. Wait, but earlier calculation showed that AB = CD and AD = BC. So maybe it's a parallelogram? Wait, in a parallelogram, opposite sides are equal and parallel. But AB and CD are equal but not parallel (slopes 1 and -1). AD and BC are equal but not parallel (slopes -5/3 and 5/3). Therefore, not a parallelogram. Therefore, my previous conclusion that it's a kite is wrong? Wait, how? If AB = CD and AD = BC, then it's a parallelogram? Wait, no. In a parallelogram, opposite sides are equal and parallel. Here, AB and CD are equal but not parallel. AD and BC are equal but not parallel. Therefore, it's not a parallelogram. So maybe it's a different type of quadrilateral. Then why did I think it's a kite? Because two pairs of adjacent sides are equal? Wait, in a kite, two distinct pairs of adjacent sides are equal. In this case, AB = CD and AD = BC. But AB and AD are adjacent sides? Wait, AB is adjacent to AD? Yes. So AB = CD and AD = BC. But in a kite, the two pairs of adjacent sides are equal. However, in this case, AB and AD are adjacent sides, but AB ≠ AD. Wait, AB is 3√2, AD is sqrt(34). So they are not equal. Therefore, this is not a kite. Therefore, my mistake earlier. So the quadrilateral is not a kite. Therefore, the area calculation via shoelace formula is more reliable. But why did I get 15? Let me check again.

Wait, when I used the shoelace formula with the order A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A, I got total sum -30, absolute value 30, divided by 2 is 15. But if I think of the coordinates, the figure seems to have a larger area. Let me try to compute the area by breaking it into two triangles. For example, divide the quadrilateral into triangle ABC and triangle ACD? Wait, but need to make sure the triangles don't overlap. Alternatively, divide into triangle ABD and triangle BCD? Let me try.

First, triangle ABD: points A(-1,1), B(2,4), D(2,-4). Area can be calculated using determinant formula:

Area = 1/2 | (x_A(y_B - y_D) + x_B(y_D - y_A) + x_D(y_A - y_B)) |

Plugging in:

1/2 | (-1)(4 - (-4)) + 2*(-4 - 1) + 2*(1 - 4) | = 1/2 | (-1)(8) + 2*(-5) + 2*(-3) | = 1/2 | -8 -10 -6 | = 1/2 | -24 | = 12.

Then triangle BCD: points B(2,4), C(-1,-1), D(2,-4). Area:

1/2 | x_B(y_C - y_D) + x_C(y_D - y_B) + x_D(y_B - y_C) |.

Plugging in:

1/2 | 2*(-1 - (-4)) + (-1)*(-4 - 4) + 2*(4 - (-1)) | = 1/2 | 2*(3) + (-1)*(-8) + 2*(5) | = 1/2 | 6 + 8 + 10 | = 1/2 * 24 = 12.

Total area: 12 + 12 = 24. Wait, but this contradicts the shoelace formula result of 15. Which is correct? There must be an error here. Wait, if I split the quadrilateral into triangles ABD and BCD, but depending on how the quadrilateral is shaped, maybe these triangles overlap or something? Wait, if the quadrilateral is A(-1,1), B(2,4), D(2,-4), C(-1,-1), then splitting into ABD and BCD would cover the entire quadrilateral? Let me visualize. From A to B to D to C to A. If I split into ABD and BCD, then yes, they should cover the quadrilateral. But why different results? Let me check the area of triangle ABD again.

Alternatively, use coordinates for triangle ABD. Coordinates A(-1,1), B(2,4), D(2,-4). Using shoelace formula for triangle:

List points A(-1,1), B(2,4), D(2,-4), back to A(-1,1).

Compute terms:

(-1)(4) - 2(1) = -4 - 2 = -6

2*(-4) - 2*4 = -8 - 8 = -16

2*1 - (-1)*(-4) = 2 - 4 = -2

Sum: -6 -16 -2 = -24. Absolute value / 2: 12. Correct. Similarly for triangle BCD: points B(2,4), C(-1,-1), D(2,-4). Shoelace formula:

2*(-1) - (-1)*4 = -2 + 4 = 2

(-1)*(-4) - 2*(-1) = 4 + 2 = 6

2*4 - 2*(-4) = 8 + 8 = 16

Wait, no, let me do it properly. Shoelace formula for triangle BCD:

Coordinates B(2,4), C(-1,-1), D(2,-4), back to B(2,4).

First term: x_B * y_C - x_C * y_B = 2*(-1) - (-1)*4 = -2 + 4 = 2

Second term: x_C * y_D - x_D * y_C = (-1)*(-4) - 2*(-1) = 4 + 2 = 6

Third term: x_D * y_B - x_B * y_D = 2*4 - 2*(-4) = 8 + 8 = 16

Sum: 2 + 6 + 16 = 24. Absolute value / 2: 12. So total area 12 + 12 = 24. But earlier shoelace formula gave 15. There's a contradiction here. Which is correct?

Wait, maybe the order of the points in the shoelace formula was wrong. When I did the shoelace formula for the quadrilateral A, B, D, C, I got -30, which gave 15. But splitting into two triangles gives 24. Which is correct? Let me check with coordinates.

Alternatively, use vector cross product. The area of quadrilateral can be calculated as the sum of the areas of triangles ABD and BCD. If that gives 24, then that's the correct area. But why does the shoelace formula give 15? Because the order of the points was incorrect, leading to overlapping or incorrect traversal. Let me try again with the shoelace formula, making sure the points are ordered correctly.

Let me order the points in a clockwise or counter-clockwise order. Let me try A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A(-1,1). Let me plot these points:

Starting at A(-1,1), moving to B(2,4): upper right.

Then to D(2,-4): straight down along x=2 to (2,-4).

Then to C(-1,-1): moving left and up to (-1,-1).

Then back to A(-1,1): up along x=-1.

This path seems to create a quadrilateral that is a non-intersecting polygon. Let me apply shoelace formula again carefully.

List the coordinates:

A(-1,1)

B(2,4)

D(2,-4)

C(-1,-1)

A(-1,1)

Compute shoelace sum:

First, multiply xi * yi+1:

(-1)*4 + 2*(-4) + 2*(-1) + (-1)*1 = (-4) + (-8) + (-2) + (-1) = -15

Then multiply yi * xi+1:

1*2 + 4*2 + (-4)*(-1) + (-1)*(-1) = 2 + 8 + 4 + 1 = 15

Subtract the two: | -15 - 15 | = | -30 | = 30. Divide by 2: 15. So area is 15. But splitting into triangles gives 24. There's a contradiction. Which is correct?

Wait, maybe the problem is that when I split into triangles ABD and BCD, I'm actually calculating the area of a different quadrilateral? Wait, if the quadrilateral is A, B, D, C, then splitting into ABD and BCD should be correct. But why discrepancy?

Wait, let me check coordinates again. If I plot the points:

A(-1,1), B(2,4), D(2,-4), C(-1,-1). Connecting A to B to D to C to A. This creates a quadrilateral that is a bowtie? Wait, no. Wait, connecting A(-1,1) to B(2,4) to D(2,-4) to C(-1,-1) to A. Let me imagine this. From A(-1,1) to B(2,4): upper right. Then to D(2,-4): straight down. Then to C(-1,-1): left and up. Then back to A(-1,1): up. This seems to create a quadrilateral that is a trapezoid? Or maybe not. Wait, if I imagine this, the sides AB and DC are not parallel, but the sides AD and BC are not parallel. However, when I split into triangles ABD and BCD, I get 24, but shoelace formula gives 15. There must be an error in one of the methods.

Wait, let me check the shoelace formula again. Maybe I made a mistake in the calculation. Let me redo it step by step.

Shoelace formula for quadrilateral A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A(-1,1):

List the coordinates:

x: -1, 2, 2, -1, -1

y: 1, 4, -4, -1, 1

Compute sum of xi * yi+1:

(-1)*4 + 2*(-4) + 2*(-1) + (-1)*1 = (-4) + (-8) + (-2) + (-1) = -15

Compute sum of yi * xi+1:

1*2 + 4*2 + (-4)*(-1) + (-1)*(-1) = 2 + 8 + 4 + 1 = 15

Subtract: | -15 - 15 | = 30. Area = 30 / 2 = 15. So shoelace formula says 15.

But splitting into triangles gives 24. Where is the mistake?

Wait, maybe the triangles overlap? If I split the quadrilateral into ABD and BCD, maybe they overlap? Let me check. The triangle ABD is formed by points A, B, D. The triangle BCD is formed by points B, C, D. These two triangles share the side BD. So together they form the quadrilateral A, B, D, C. So they should not overlap. Therefore, their areas should add up. But why discrepancy?

Wait, let me recalculate the area of triangle ABD. Points A(-1,1), B(2,4), D(2,-4). Using shoelace formula:

Coordinates:

A(-1,1), B(2,4), D(2,-4), back to A(-1,1).

Compute sum of xi*yi+1:

(-1)*4 + 2*(-4) + 2*1 = -4 -8 + 2 = -10

Sum of yi*xi+1:

1*2 + 4*2 + (-4)*(-1) = 2 + 8 + 4 = 14

Subtract: |-10 - 14| = 24. Area = 24 / 2 = 12. Correct.

Triangle BCD: points B(2,4), C(-1,-1), D(2,-4). Shoelace formula:

Coordinates:

B(2,4), C(-1,-1), D(2,-4), back to B(2,4).

Sum of xi*yi+1:

2*(-1) + (-1)*(-4) + 2*4 = -2 + 4 + 8 = 10

Sum of yi*xi+1:

4*(-1) + (-1)*2 + (-4)*2 = -4 -2 -8 = -14

Subtract: |10 - (-14)| = |24|, area = 24 / 2 = 12. So total area 12 + 12 = 24. But shoelace formula for quadrilateral gives 15. This is a contradiction. There must be a mistake in the order of the points in the shoelace formula for the quadrilateral. Maybe the order is not correct, leading to the shoelace formula calculating the area of a different polygon?

Wait, if I take the order A, B, D, C, the shoelace formula gives 15, but splitting into triangles gives 24. Let me try another order. What if I take the order A, B, C, D? Let me try that.

Order: A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A(-1,1).

Compute shoelace formula:

Sum of xi*yi+1:

(-1)*4 + 2*(-1) + (-1)*(-4) + 2*1 = -4 -2 + 4 + 2 = 0

Sum of yi*xi+1:

1*2 + 4*(-1) + (-1)*2 + (-4)*(-1) = 2 -4 -2 + 4 = 0

Subtract: |0 - 0| = 0. Area 0. Which is wrong. So this order is incorrect.

Another order: A, C, B, D.

Order: A(-1,1), C(-1,-1), B(2,4), D(2,-4), back to A.

Compute shoelace:

Sum xi*yi+1:

(-1)*(-1) + (-1)*4 + 2*(-4) + 2*1 = 1 -4 -8 + 2 = -9

Sum yi*xi+1:

1*(-1) + (-1)*2 + 4*2 + (-4)*(-1) = -1 -2 + 8 + 4 = 9

Subtract: |-9 - 9| = 18. Area 9. Still not matching.

Another order: A, D, B, C.

Order: A(-1,1), D(2,-4), B(2,4), C(-1,-1), back to A.

Compute:

Sum xi*yi+1:

(-1)*(-4) + 2*4 + 2*(-1) + (-1)*1 = 4 + 8 - 2 -1 = 9

Sum yi*xi+1:

1*2 + (-4)*2 + 4*(-1) + (-1)*(-1) = 2 -8 -4 + 1 = -9

Subtract: |9 - (-9)| = 18. Area 9. Still not matching.

Wait, so why does splitting into triangles give 24, but shoelace formula with different orders gives 15 or 9 or 0? It seems that the correct area is 24, and the shoelace formula was giving wrong results because of incorrect ordering. How to determine the correct order?

Let me think of the quadrilateral as a non-intersecting polygon. If I traverse the points in the order A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A, does this create a non-intersecting polygon? Let me imagine:

Start at A(-1,1), go to B(2,4), then to D(2,-4), then to C(-1,-1), then back to A. This path would cross over itself? Let me check. From A to B to D: that's a line from A to B to D. Then from D to C to A: from D(2,-4) to C(-1,-1) to A(-1,1). This seems to create a quadrilateral that is a non-intersecting polygon. But why does the shoelace formula give 15? Maybe I made a mistake in the shoelace calculation. Let me try once more with this order.

Order: A(-1,1), B(2,4), D(2,-4), C(-1,-1), A(-1,1).

Compute shoelace formula step by step:

List the coordinates:

x: -1, 2, 2, -1, -1

y: 1, 4, -4, -1, 1

Compute sum of xi * yi+1:

(-1)*4 + 2*(-4) + 2*(-1) + (-1)*1 = (-4) + (-8) + (-2) + (-1) = -15

Compute sum of yi * xi+1:

1*2 + 4*2 + (-4)*(-1) + (-1)*(-1) = 2 + 8 + 4 + 1 = 15

Subtract: |-15 - 15| = 30. Area = 30 / 2 = 15.

But splitting into triangles gives 24. This is very confusing. Let me try to calculate the area using coordinates with vectors or integration.

Alternatively, think of the quadrilateral as a polygon with vertices at (-1,1), (2,4), (2,-4), (-1,-1). Let me use the shoelace formula with a different order that ensures the polygon is traversed correctly. Let me try the order A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A. Wait, earlier this gave zero. But maybe I did it wrong. Let me recalculate:

Order: A(-1,1), B(2,4), C(-1,-1), D(2,-4), A(-1,1).

Compute xi*yi+1:

(-1)*4 + 2*(-1) + (-1)*(-4) + 2*1 = -4 -2 + 4 + 2 = 0

Compute yi*xi+1:

1*2 + 4*(-1) + (-1)*2 + (-4)*(-1) = 2 -4 -2 + 4 = 0

Subtract: 0 - 0 = 0. Area 0. This is wrong. So this order is incorrect.

Wait, maybe the correct order is A(-1,1), B(2,4), D(2,-4), C(-1,-1), A(-1,1). But why does this give 15? Let me try to visualize the area. If I imagine the quadrilateral, it's like a bowtie? No, because it's non-intersecting. Wait, if I connect A to B to D to C to A, it should be a quadrilateral. Let me calculate the area using coordinates with vectors.

Alternatively, use the shoelace formula with the correct order. Maybe I need to ensure that the points are ordered either clockwise or counter-clockwise without crossing.

Let me try ordering the points as A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A. Let me plot this:

Start at A(-1,1), go to B(2,4): upper right.

From B(2,4) to D(2,-4): straight down along x=2.

From D(2,-4) to C(-1,-1): left and up.

From C(-1,-1) back to A(-1,1): up along x=-1.

This should form a quadrilateral that is a trapezoid? Wait, no. Wait, if you imagine this, the sides AB and DC are not parallel, but the sides AD and BC are not parallel. However, the area calculated by shoelace is 15, while splitting into triangles gives 24. This is very confusing. Maybe I need to check with coordinates.

Let me calculate the area using the shoelace formula with a different approach. Let me use the coordinates and calculate the area step by step.

The shoelace formula is sensitive to the order of the points. If the points are ordered incorrectly, it can give wrong results. Let me try to order the points in a counter-clockwise manner.

Let me list the points:

A(-1,1), B(2,4), D(2,-4), C(-1,-1).

To order them counter-clockwise, let's imagine the positions:

A is on the left upper, B is on the right upper, D is on the right lower, C is on the left lower.

A counter-clockwise order would be A, B, D, C. Let me check:

Starting at A(-1,1), moving to B(2,4) (upper right), then to D(2,-4) (down), then to C(-1,-1) (left and up), then back to A. This seems counter-clockwise? Or is it clockwise? It's hard to tell without a diagram. Let me use the shoelace formula with this order and see.

As before, it gives 15. But splitting into triangles gives 24. There must be a mistake in one of the methods. Let me try to calculate the area using coordinates with vectors.

The area can also be calculated using the vector cross product. For a polygon, the area is 1/2 |sum_{i=1 to n} (xi*yi+1 - xi+1*yi)|. So if I trust the shoelace formula, it's 15. But why does splitting into triangles give 24? Let me check the coordinates again.

Wait, if I take the quadrilateral A(-1,1), B(2,4), D(2,-4), C(-1,-1), and calculate the area using the shoelace formula, it's 15. But if I calculate the area of the triangle ABD (12) and BCD (12), it's 24. This suggests that the shoelace formula is giving the area of a different quadrilateral. Perhaps the order of the points in the shoelace formula is creating a self-intersecting polygon, which results in subtracting areas?

Wait, if the order A, B, D, C creates a self-intersecting polygon, then the shoelace formula would calculate the net area, which could be less than the actual area. Let me check if the polygon is self-intersecting. From A(-1,1) to B(2,4) to D(2,-4) to C(-1,-1) to A(-1,1). Does this cross over? Let me see: the line from B(2,4) to D(2,-4) is vertical. The line from D(2,-4) to C(-1,-1) goes from (2,-4) to (-1,-1). The line from C(-1,-1) to A(-1,1) is vertical. The line from A(-1,1) to B(2,4) is diagonal. I don't see any intersections. So it's a simple quadrilateral. Therefore, the shoelace formula should give the correct area. But why discrepancy with the triangle areas?

Wait, let me calculate the area of the quadrilateral using another method. Let me use coordinates to integrate or use the formula for the area of a trapezoid. Wait, but it's not a trapezoid. Alternatively, use coordinates to find the area.

Let me consider the coordinates of the quadrilateral. It has vertices at (-1,1), (2,4), (2,-4), (-1,-1). Let me plot these points on a coordinate system.

Point A(-1,1) is on the left side, upper.

Point B(2,4) is on the right side, upper.

Point D(2,-4) is on the right side, lower.

Point C(-1,-1) is on the left side, lower.

Connecting these points in order A-B-D-C-A. This creates a quadrilateral that is symmetric about the x-axis? Wait, no. Wait, point A(-1,1) and C(-1,-1) are symmetric about the x-axis. Point B(2,4) and D(2,-4) are symmetric about the x-axis. So the entire quadrilateral is symmetric about the x-axis. Therefore, the area should be twice the area of the upper half. Let me calculate the area of the upper half.

The upper half consists of the points A(-1,1), B(2,4), and the x-axis. Wait, no. The upper half of the quadrilateral would be the area between A(-1,1), B(2,4), and the line connecting B(2,4) to the x-axis? No, it's better to think of it as the area above the x-axis. Since the quadrilateral is symmetric, the area above the x-axis is equal to the area below. Let me calculate the area above the x-axis.

The upper part of the quadrilateral is the polygon formed by A(-1,1), B(2,4), and the intersection of the line BD with the x-axis? Wait, no. Wait, the quadrilateral is A-B-D-C-A. The part above the x-axis is the triangle A-B-D? No, because D is below the x-axis. Wait, this is getting too confusing. Let me use the symmetry. Since the quadrilateral is symmetric about the x-axis, the area should be twice the area of the part above the x-axis. The part above the x-axis is the area bounded by A(-1,1), B(2,4), and the x-axis? No, it's bounded by A(-1,1), B(2,4), and the line from B(2,4) to D(2,-4) which crosses the x-axis at (2,0). Wait, this is getting too complex. Let me instead calculate the area of the upper half by integrating or using coordinates.

Alternatively, since the quadrilateral is symmetric, let me calculate the area of the upper half, which is the area between A(-1,1), B(2,4), and the x-axis. Wait, but how? The upper half of the quadrilateral would be the area bounded by A(-1,1), B(2,4), and the line from B(2,4) to the x-axis? No, I need to define it properly.

Alternatively, since the quadrilateral is symmetric, the area above the x-axis is the area of the triangle formed by A(-1,1), B(2,4), and the projection of B on the x-axis, which is (2,0). Similarly, the area below the x-axis is the same. But this might not be accurate.

Let me calculate the area above the x-axis. It would be the area of the polygon A(-1,1), B(2,4), (2,0), and back to A(-1,1)? No, not sure. This is getting too complicated. Let me go back to the shoelace formula. If the shoelace formula with the correct order gives 15, and splitting into triangles gives 24, there must be a mistake in one of the calculations. Let me check the triangle areas again.

For triangle ABD: points A(-1,1), B(2,4), D(2,-4). Using the formula for area of a triangle with coordinates:

Area = 1/2 | (x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) |

Plugging in:

1/2 | (-1)(4 - (-4)) + 2(-4 - 1) + 2(1 - 4) | = 1/2 | (-1)(8) + 2(-5) + 2(-3) | = 1/2 | -8 -10 -6 | = 1/2 | -24 | = 12. Correct.

For triangle BCD: points B(2,4), C(-1,-1), D(2,-4). Using the same formula:

1/2 | 2(-1 - (-4)) + (-1)(-4 - 4) + 2(4 - (-1)) | = 1/2 | 2(3) + (-1)(-8) + 2(5) | = 1/2 | 6 + 8 + 10 | = 1/2 * 24 = 12. Correct.

So total area 24. But why does the shoelace formula give 15? It must be that the order of the points in the shoelace formula is incorrect, leading to the calculation of a different area. Let me try to find the correct order.

Let me try ordering the points as A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A. Wait, this gave zero. Let me try another order: A(-1,1), D(2,-4), C(-1,-1), B(2,4), back to A. Let's apply shoelace formula.

Coordinates: A(-1,1), D(2,-4), C(-1,-1), B(2,4), A(-1,1).

Compute xi*yi+1:

(-1)*(-4) + 2*(-1) + (-1)*4 + 2*1 = 4 - 2 - 4 + 2 = 0

Compute yi*xi+1:

1*2 + (-4)*(-1) + (-1)*2 + 4*(-1) = 2 + 4 - 2 - 4 = 0

Subtract: 0 - 0 = 0. Area 0. Still wrong.

Wait, maybe the correct order is A(-1,1), B(2,4), D(2,-4), C(-1,-1), A(-1,1). Let me try to visualize this as a non-intersecting polygon. If I go from A to B to D to C to A, it should be a quadrilateral. Let me calculate the area using the shoelace formula with this order, but this time, I'll use the formula step by step.

List of points:

1. A(-1,1)
2. B(2,4)
3. D(2,-4)
4. C(-1,-1)
5. A(-1,1)

Compute the shoelace sum:

Sum1 = (x1*y2 + x2*y3 + x3*y4 + x4*y5)

= (-1)*4 + 2*(-4) + 2*(-1) + (-1)*1 = -4 -8 -2 -1 = -15

Sum2 = (y1*x2 + y2*x3 + y3*x4 + y4*x5)

= 1*2 + 4*2 + (-4)*(-1) + (-1)*(-1) = 2 + 8 + 4 + 1 = 15

Area = 1/2 |Sum1 - Sum2| = 1/2 |-15 - 15| = 1/2 * 30 = 15.

But why is this different from the triangle method? It must be that the shoelace formula is calculating the area of a different polygon. Wait, if I connect A to B to D to C to A, it's possible that this creates a self-intersecting polygon, but I don't see how. Let me check if the lines cross. The line from B(2,4) to D(2,-4) is vertical. The line from D(2,-4) to C(-1,-1) is from (2,-4) to (-1,-1). The line from C(-1,-1) to A(-1,1) is vertical. The line from A(-1,1) to B(2,4) is diagonal. No intersections. So it's a simple quadrilateral. Therefore, the area should be 15. But why does splitting into triangles give 24? This is very confusing.

Wait, let me calculate the area using the shoelace formula with the correct order, and also calculate the area using the triangle method, but with the correct triangles. Maybe I chose the wrong triangles. Let me think: if the quadrilateral is A-B-D-C, then the correct way to split it into triangles is A-B-D and A-D-C? Or A-B-C and B-C-D? Wait, no. If the quadrilateral is A-B-D-C, then the diagonals are A-D and B-C. So splitting into triangles A-B-D and A-D-C. Let me try that.

Triangle A-B-D: area 12.

Triangle A-D-C: points A(-1,1), D(2,-4), C(-1,-1). Let me calculate this area.

Using the formula:

1/2 | x_A(y_D - y_C) + x_D(y_C - y_A) + x_C(y_A - y_D) |

= 1/2 | (-1)(-4 - (-1)) + 2(-1 - 1) + (-1)(1 - (-4)) |

= 1/2 | (-1)(-3) + 2(-2) + (-1)(5) |

= 1/2 | 3 - 4 - 5 | = 1/2 | -6 | = 3.

So total area would be 12 + 3 = 15. Ah! Now it matches the shoelace formula. So earlier, I was splitting into triangles ABD and BCD, which were not part of the quadrilateral A-B-D-C. Instead, the correct split is ABD and ADC, giving 12 + 3 = 15. So my mistake was in choosing the wrong triangles. Earlier, I split into ABD and BCD, but BCD is not part of the quadrilateral A-B-D-C. The correct split is ABD and ADC, giving 15. Therefore, the correct area is 15.

So the area of the quadrilateral is 15. But let me confirm with the shoelace formula. Yes, it gave 15. So the correct area is 15. Therefore, the answer is 15. But let me check with the Asymptote code. The Asymptote code draws points A(-1,1), B(2,4), C(-1,-1), D(2,-4), and the area is 15. But the problem says "the area of the quadrilateral formed by the four points of tangency." So the answer is 15? But let me check with the original problem.

Wait, the original problem says the circle x² + y² = 2 and the parabola y² = 8x have two common tangents, forming four points of tangency. Find the area of the quadrilateral formed by the four points of tangency.

But according to our calculation, the area is 15. But let me check if the points A(-1,1), B(2,4), C(-1,-1), D(2,-4) are indeed the points of tangency. Earlier, we checked that the line AB is tangent to both the circle and the parabola, and similarly for CD. So these are the points of tangency. Therefore, the quadrilateral formed by these four points has area 15. But let me check if this is correct.

Wait, but the circle has radius sqrt(2) ≈ 1.414, and the points A(-1,1) and C(-1,-1) are on the circle. Let me check the distance from the origin to A: sqrt((-1)^2 + 1^2) = sqrt(2), correct. Similarly for C. Points B(2,4) and D(2,-4) are on the parabola y² = 8x: 4² = 16, 8*2 = 16, correct. So these are indeed the points of tangency. Therefore, the area is 15. But let me check with another method.

Alternatively, since the quadrilateral is symmetric about the x-axis, we can calculate the area of the upper half and double it. The upper half is the area between A(-1,1), B(2,4), and the x-axis. Wait, but how? Let me calculate the area of the upper part as the area of the triangle ABD, but no. Wait, the upper part is the area bounded by A(-1,1), B(2,4), and the line from B(2,4) to the x-axis. Let me calculate this.

The line from B(2,4) to the x-axis is the vertical line x=2 from (2,4) to (2,0). The area of the upper part would be the area of the trapezoid formed by A(-1,1), B(2,4), (2,0), and the projection of A on the x-axis, which is (-1,0). But this is getting complex. Let me calculate it.

The area of the upper part can be calculated as the area of the trapezoid with vertices at A(-1,1), B(2,4), (2,0), and (-1,0). The area of this trapezoid is the average of the two bases times the height. The bases are the lengths from (-1,1) to (-1,0) which is 1, and from (2,4) to (2,0) which is 4. The height is the horizontal distance between x=-1 and x=2, which is 3. Wait, no. The formula for the area of a trapezoid is (base1 + base2)/2 * height, where height is the distance between the two bases. In this case, the two bases are the vertical segments at x=-1 and x=2, with lengths 1 and 4. The distance between these bases is the horizontal distance between x=-1 and x=2, which is 3. So area is (1 + 4)/2 * 3 = 5/2 * 3 = 15/2. Then the total area would be twice this, 15. Which matches the previous result. Therefore, the area is indeed 15.

Therefore, the answer is 15. But let me check if the problem might expect a different answer. Wait, the Asymptote code in the problem has points A(-1,1), B(2,4), C(-1,-1), D(2,-4), and draws lines connecting A to B, C to D, and A to C, B to D. The area of the quadrilateral formed by these points is 15. Therefore, the answer should be 15. But let me check if there is a calculation error.

Wait, the trapezoid area calculation gave 15/2 for the upper half, so total area 15. Yes. And the shoelace formula and triangle split (correctly) also gave 15. Therefore, the correct area is 15. So the answer is 15, which in boxed form is \boxed{15}.

**Final Answer**
\boxed{15}

To find the area of the quadrilateral formed by the four points of tangency between the circle $ x^2 + y^2 = 2 $ and the parabola $ y^2 = 8x $, we begin by identifying the points of tangency.

---

### Step 1: Identify the Points of Tangency

We are told that the circle and the parabola have **two common tangents**, each of which touches both the circle and the parabola at one point. Thus, there are **four points of tangency** in total: two on the circle and two on the parabola.

From the Asymptote code and the problem's context, the points of tangency are:

- $ A = (-1, 1) $ — on the circle
- $ B = (2, 4) $ — on the parabola
- $ C = (-1, -1) $ — on the circle
- $ D = (2, -4) $ — on the parabola

These points form a quadrilateral $ ABCD $, which is symmetric about the x-axis.

---

### Step 2: Use the Shoelace Formula to Compute the Area

We apply the **shoelace formula** to compute the area of quadrilateral $ ABCD $, using the coordinates in the order $ A \to B \to D \to C \to A $:

$$
\text{Area} = \frac{1}{2} \left| \sum (x_i y_{i+1} - x_{i+1} y_i) \right|
$$

List of points in order:
- $ A = (-1, 1) $
- $ B = (2, 4) $
- $ D = (2, -4) $
- $ C = (-1, -1) $
- Back to $ A = (-1, 1) $

Compute the terms:

- $ x_A y_B = (-1)(4) = -4 $
- $ x_B y_D = (2)(-4) = -8 $
- $ x_D y_C = (2)(-1) = -2 $
- $ x_C y_A = (-1)(1) = -1 $

Sum of $ x_i y_{i+1} $:
$$
-4 - 8 - 2 - 1 = -15
$$

Now compute the other sum:

- $ y_A x_B = (1)(2) = 2 $
- $ y_B x_D = (4)(2) = 8 $
- $ y_D x_C = (-4)(-1) = 4 $
- $ y_C x_A = (-1)(-1) = 1 $

Sum of $ y_i x_{i+1} $:
$$
2 + 8 + 4 + 1 = 15
$$

Now compute the area:

$$
\text{Area} = \frac{1}{2} | -15 - 15 | = \frac{1}{2} \cdot 30 = 15
$$

---

### Step 3: Confirm with Triangle Decomposition

To verify, we can split the quadrilateral into two triangles:

- Triangle $ ABD $: Points $ A(-1,1), B(2,4), D(2,-4) $
- Triangle $ ACD $: Points $ A(-1,1), C(-1,-1), D(2,-4) $

Using the shoelace formula for each triangle:

- Area of $ ABD $: $ \frac{1}{2} | -24 | = 12 $
- Area of $ ACD $: $ \frac{1}{2} | -6 | = 3 $

Total area: $ 12 + 3 = 15 $

---

### Final Answer

$$
\boxed{15}
$$

[Assistant]
Explanation:
Okay, so I need to find the area of the quadrilateral formed by the four points of tangency between the circle x² + y² = 2 and the parabola y² = 8x. The problem mentions there are two common tangents, which form four points of tangency. Wait, hold on, two common tangents? But tangents usually touch at one point each. If there are two common tangents, each tangent would have one point of tangency, so that would be two points. But the problem says four points of tangency. Maybe I need to clarify that.

Wait, maybe the circle and the parabola have two common tangents, but each tangent touches both the circle and the parabola? So each tangent line is tangent to both the circle and the parabola. Therefore, each tangent line would have one point of tangency on the circle and one on the parabola. Therefore, two common tangents would result in two points on the circle and two points on the parabola? But the problem says four points of tangency. Maybe there are two common tangents, each tangent to both the circle and the parabola, so each tangent has two points of tangency? Wait, no. A tangent line to a curve touches it at one point. So if there are two common tangents, each tangent line touches the circle at one point and the parabola at another point. Therefore, there are two points on the circle and two points on the parabola. But the problem says four points of tangency. Maybe the problem is referring to the four points where the common tangents touch the circle and the parabola? So two tangents, each touching the circle once and the parabola once, so total four points. Then the quadrilateral is formed by these four points. So the four points are two on the circle and two on the parabola? Or maybe the tangents are such that each tangent touches both curves at the same point? But that seems unlikely unless they intersect at a point where they are tangent to both. But the circle and parabola might intersect? Let me check.

First, let me check if the circle and parabola intersect. The circle is x² + y² = 2, and the parabola is y² = 8x. Substitute y² from the parabola into the circle equation: x² + 8x = 2 => x² + 8x - 2 = 0. The discriminant is 64 + 8 = 72, so there are two real solutions. Therefore, the circle and parabola intersect at two points. Therefore, they are not tangent to each other. Therefore, the common tangents are lines that touch both the circle and the parabola without intersecting them. Therefore, each common tangent touches the circle at one point and the parabola at another point. Therefore, there are two common tangents, each contributing one point on the circle and one on the parabola, so four points total. Then the quadrilateral is formed by connecting these four points. Wait, but how? If there are two points on the circle and two on the parabola, how do they form a quadrilateral? Maybe the four points are two on the circle and two on the parabola, but arranged such that they form a quadrilateral. But maybe the problem is referring to the points where the common tangents touch the curves. So each tangent line touches the circle and the parabola, so each tangent line has two points of tangency. Therefore, two tangents would have four points of tangency. Therefore, the quadrilateral is formed by connecting these four points. So the four points are two on the circle and two on the parabola. But how do they form a quadrilateral? Maybe the two tangents are symmetric with respect to the x-axis? Let me think.

Alternatively, maybe there are four common tangents? Wait, the problem says two common tangents. Wait, maybe the Asymptote code gives some clue. The Asymptote code draws the upper and lower parabola branches, the circle, and some lines. The points A, B, C, D are given as (-1,1), (2,4), (-1,-1), (2,-4). Then it draws the graph of the upper and lower parabola, the circle, and some lines connecting A to B and C to D, and also A to C and B to D. The points A, B, C, D are probably the points of tangency. So maybe the quadrilateral is A, B, C, D? Let me check coordinates. If A is (-1,1), then plugging into the circle equation: (-1)^2 + (1)^2 = 1 + 1 = 2, which satisfies the circle equation. Similarly, point C is (-1,-1), which also satisfies the circle equation. Then points B (2,4): 2² + 4² = 4 + 16 = 20 ≠ 2. Wait, that's not on the circle. Wait, but the problem says the quadrilateral is formed by the four points of tangency. But according to the Asymptote code, points A and C are on the circle, but points B and D are not. So maybe the Asymptote code is illustrative but not exact? Or maybe the coordinates given in the Asymptote code are not the actual points of tangency? Because if points B and D are on the parabola, let's check: for point B (2,4), y² = 16, and 8x = 16, so yes, that's on the parabola. Similarly, D (2,-4) is also on the parabola. So points A and C are on the circle, points B and D are on the parabola. Then the quadrilateral is formed by connecting A, B, C, D? But how? If you connect A to B to C to D to A? Let me visualize. A is (-1,1), B is (2,4), C is (-1,-1), D is (2,-4). So connecting A(-1,1) to B(2,4), then to C(-1,-1), then to D(2,-4), then back to A? That seems like a quadrilateral. But maybe the actual points of tangency are different? Wait, but the Asymptote code is just an illustration. The problem says "the quadrilateral formed by the four points of tangency," so maybe the answer is the area of quadrilateral ABCD as given? But the coordinates given in the Asymptote code are probably not the actual points of tangency. Because if you check the tangent lines from A to B, but A is on the circle and B is on the parabola. Wait, but the problem says the common tangents. So each tangent line is tangent to both the circle and the parabola. Therefore, each tangent line touches the circle at one point and the parabola at another. Therefore, there are two such tangent lines, each contributing two points of tangency. Therefore, four points total. So the quadrilateral is formed by these four points. Therefore, the Asymptote code's points A, B, C, D are probably the actual points of tangency. But in that case, the coordinates given in the Asymptote code are (-1,1), (2,4), (-1,-1), (2,-4). Let me check if these points are points of tangency. For example, take point A (-1,1). Is there a tangent line to both the circle and the parabola that touches the circle at A and the parabola at B (2,4)? Let me check if the line AB is tangent to both curves.

First, find the equation of the line AB. Points A(-1,1) and B(2,4). The slope is (4 - 1)/(2 - (-1)) = 3/3 = 1. So the equation is y - 1 = 1*(x + 1), so y = x + 2. Let me check if this line is tangent to the circle x² + y² = 2. Substitute y = x + 2 into the circle equation:

x² + (x + 2)^2 = 2 => x² + x² + 4x + 4 = 2 => 2x² + 4x + 2 = 0 => x² + 2x + 1 = 0 => (x + 1)^2 = 0. So x = -1, which is the point A(-1,1). Therefore, the line AB is tangent to the circle at A. Now check if it's tangent to the parabola y² = 8x. Substitute y = x + 2 into the parabola equation:

(x + 2)^2 = 8x => x² + 4x + 4 = 8x => x² - 4x + 4 = 0 => (x - 2)^2 = 0 => x = 2. Therefore, the line AB touches the parabola at x = 2, y = 4, which is point B. Therefore, the line AB is indeed a common tangent to both the circle and the parabola, touching the circle at A and the parabola at B. Similarly, the line CD connects C(-1,-1) and D(2,-4). Let me check that. The slope is (-4 - (-1))/(2 - (-1)) = (-3)/3 = -1. Equation: y - (-1) = -1(x - (-1)) => y + 1 = -x -1 => y = -x - 2. Check if this is tangent to the circle. Substitute y = -x - 2 into x² + y² = 2:

x² + (-x - 2)^2 = 2 => x² + x² + 4x + 4 = 2 => 2x² + 4x + 2 = 0 => same as before, x = -1. Therefore, tangent at point C(-1,-1). Check if tangent to parabola y² = 8x. Substitute y = -x - 2:

(-x - 2)^2 = 8x => x² + 4x + 4 = 8x => x² - 4x + 4 = 0 => (x - 2)^2 = 0 => x = 2. Therefore, tangent at point D(2,-4). Therefore, the line CD is also a common tangent. Therefore, the four points of tangency are A(-1,1), B(2,4), C(-1,-1), D(2,-4). Therefore, the quadrilateral is formed by these four points. Therefore, the area of quadrilateral ABCD is the area we need to find. However, the Asymptote code also draws lines connecting A to B, C to D, and A to C, B to D. So the quadrilateral is probably a trapezoid or some quadrilateral. Let me plot these points mentally. Points A(-1,1), B(2,4), C(-1,-1), D(2,-4). So plotting these, A is on the left upper, B is on the right upper, C is on the left lower, D is on the right lower. Connecting A to B to C to D to A. This seems like a trapezoid? Let me check if sides AB and CD are parallel. The slope of AB is 1, slope of CD is -1. Not parallel. The slope of BC: from B(2,4) to C(-1,-1): slope is (-1 - 4)/(-1 - 2) = (-5)/(-3) = 5/3. Slope of DA: from D(2,-4) to A(-1,1): slope is (1 - (-4))/(-1 - 2) = 5/(-3) = -5/3. Not parallel. So maybe it's a kite? Or some other quadrilateral. Alternatively, maybe it's a parallelogram? Wait, no. Alternatively, maybe it's symmetric with respect to the x-axis. Let me check coordinates. Points A(-1,1) and C(-1,-1) are symmetric over the x-axis. Similarly, points B(2,4) and D(2,-4) are symmetric over the x-axis. Therefore, the quadrilateral is symmetric over the x-axis. Therefore, maybe we can compute the area by computing the area of the upper half and doubling it. Let me think. Alternatively, since it's symmetric, maybe we can use coordinates to compute the area.

Alternatively, since we have coordinates of all four points, we can use the shoelace formula. Let me recall the shoelace formula. For a polygon with vertices (x1,y1), (x2,y2), ..., (xn,yn), the area is |1/2 * sum_{i=1 to n} (xi*yi+1 - xi+1*yi)|, where xn+1=x1, yn+1=y1.

So let's apply that. Let me list the coordinates in order. Let's take A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A(-1,1).

Compute the terms:

x1*y2 - x2*y1: (-1)*4 - 2*1 = -4 - 2 = -6

x2*y3 - x3*y2: 2*(-1) - (-1)*4 = -2 + 4 = 2

x3*y4 - x4*y3: (-1)*(-4) - 2*(-1) = 4 + 2 = 6

x4*y1 - x1*y4: 2*1 - (-1)*(-4) = 2 - 4 = -2

Sum these up: -6 + 2 + 6 - 2 = 0? Wait, that can't be. Wait, maybe I did the shoelace formula wrong. Let me check again.

Wait, the shoelace formula is sum over (xi * yi+1 - xi+1 * yi). So let me write down all terms step by step.

Order of points: A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A(-1,1).

Compute each term:

First term: x_A * y_B - x_B * y_A = (-1)(4) - (2)(1) = -4 - 2 = -6

Second term: x_B * y_C - x_C * y_B = (2)(-1) - (-1)(4) = -2 + 4 = 2

Third term: x_C * y_D - x_D * y_C = (-1)(-4) - (2)(-1) = 4 + 2 = 6

Fourth term: x_D * y_A - x_A * y_D = (2)(1) - (-1)(-4) = 2 - 4 = -2

Sum of all terms: -6 + 2 + 6 - 2 = 0. Wait, that gives zero? That can't be. That would mean the area is zero, which is impossible. So I must have messed up the order of the points. Maybe the points are not ordered correctly for the shoelace formula. Let me check the order. If I connect A(-1,1) to B(2,4) to C(-1,-1) to D(2,-4) to A(-1,1), is that a correct order? Let me visualize again. If you go from A(-1,1) to B(2,4), then to C(-1,-1), that would cross over? Maybe the order is not correct. Maybe I need to order the points in a cyclic order around the quadrilateral. Let me think. If the quadrilateral is symmetric over the x-axis, maybe the correct order is A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A. Let me try that.

Order: A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A(-1,1).

Compute terms:

First term: x_A * y_B - x_B * y_A = (-1)(4) - 2(1) = -4 - 2 = -6

Second term: x_B * y_D - x_D * y_B = 2*(-4) - 2*4 = -8 - 8 = -16

Third term: x_D * y_C - x_C * y_D = 2*(-1) - (-1)*(-4) = -2 - 4 = -6

Fourth term: x_C * y_A - x_A * y_C = (-1)(1) - (-1)*(-1) = -1 - 1 = -2

Sum: -6 -16 -6 -2 = -30. Take absolute value and divide by 2: | -30 | / 2 = 15. So area is 15? Wait, but let me check again. Maybe I made a mistake in the order.

Alternatively, maybe the correct order is A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A. But that gave zero. Maybe I need to arrange the points in a different order. Let me try another order. Let me think of the quadrilateral as two triangles. For example, A, B, D, C. Wait, maybe the quadrilateral is actually a parallelogram? Wait, if we connect A(-1,1) to B(2,4) to D(2,-4) to C(-1,-1) to A. Let me check if AB is parallel to CD? Wait, AB has slope 1, CD is from C(-1,-1) to D(2,-4): slope is (-4 - (-1))/(2 - (-1)) = (-3)/3 = -1. Not parallel. AD is from A(-1,1) to D(2,-4): slope is (-4 - 1)/(2 - (-1)) = -5/3. BC is from B(2,4) to C(-1,-1): slope is (-1 - 4)/(-1 - 2) = -5/-3 = 5/3. So AD and BC have slopes -5/3 and 5/3, which are negative reciprocals? No, they are negative of each other. So if AB is slope 1, BC is 5/3, CD is -1, DA is -5/3. Not sure. Maybe it's a trapezoid? Wait, if AB and CD are not parallel, and AD and BC are not parallel. So maybe it's a general quadrilateral. Let me try another order. Maybe A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A. Let me recalculate the shoelace formula with this order.

Coordinates:

A(-1,1), B(2,4), D(2,-4), C(-1,-1), A(-1,1)

Compute terms:

First term: x_A * y_B - x_B * y_A = (-1)(4) - 2(1) = -4 - 2 = -6

Second term: x_B * y_D - x_D * y_B = 2*(-4) - 2*4 = -8 - 8 = -16

Third term: x_D * y_C - x_C * y_D = 2*(-1) - (-1)*(-4) = -2 - 4 = -6

Fourth term: x_C * y_A - x_A * y_C = (-1)(1) - (-1)*(-1) = -1 - 1 = -2

Total sum: -6 -16 -6 -2 = -30. Absolute value divided by 2: 15. So area is 15? But let me check with another order. Suppose I take A(-1,1), C(-1,-1), D(2,-4), B(2,4), back to A. Let me compute:

First term: x_A * y_C - x_C * y_A = (-1)(-1) - (-1)(1) = 1 + 1 = 2

Second term: x_C * y_D - x_D * y_C = (-1)(-4) - 2*(-1) = 4 + 2 = 6

Third term: x_D * y_B - x_B * y_D = 2*4 - 2*(-4) = 8 + 8 = 16

Fourth term: x_B * y_A - x_A * y_B = 2*1 - (-1)*4 = 2 + 4 = 6

Total sum: 2 + 6 + 16 + 6 = 30. Absolute value divided by 2: 15. Same result. So area is 15? But let me check with coordinates. Alternatively, since the quadrilateral is symmetric over the x-axis, maybe we can compute the area of the upper half and double it. The upper half would be the triangle ABD? Wait, no. Wait, points A(-1,1), B(2,4), and the reflection of D is B? Wait, no. Wait, the upper half would be the area between A(-1,1), B(2,4), and the x-axis? Maybe not. Alternatively, since the quadrilateral is symmetric over the x-axis, we can compute the area of the upper half (the part above the x-axis) and double it. The upper half would be the triangle formed by A(-1,1), B(2,4), and the projection on the x-axis? Wait, maybe not. Alternatively, think of the quadrilateral as two congruent trapezoids? Wait, maybe not. Alternatively, since we have coordinates, maybe using coordinates is the safest way. But according to shoelace formula, the area is 15. However, let me check with another method. For example, coordinates of A(-1,1), B(2,4), C(-1,-1), D(2,-4). Let me compute vectors or use coordinates.

Alternatively, since the quadrilateral is symmetric over the x-axis, we can compute the area of the upper half and double it. The upper half consists of points A(-1,1), B(2,4), and the reflection of C and D over the x-axis. Wait, but since it's symmetric, the upper half would be a triangle or a trapezoid. Wait, if we take the upper half, it's the area between A(-1,1), B(2,4), and the x-axis? Wait, no. The quadrilateral is formed by A, B, C, D. If we split it along the x-axis, we have two congruent parts. The upper part is the triangle ABD? Wait, no. Wait, if you imagine the quadrilateral, it's like a bowtie? Wait, no. Wait, if you connect A(-1,1) to B(2,4) to D(2,-4) to C(-1,-1) to A, then it's a non-intersecting quadrilateral? Wait, connecting A to B to D to C to A. Let me check if the sides cross. From A(-1,1) to B(2,4), then to D(2,-4): that's a line from B(2,4) down to D(2,-4), which is vertical line x=2. Then from D(2,-4) to C(-1,-1), then back to A(-1,1). So this seems like a quadrilateral that is a trapezoid? Wait, but with one vertical side? Wait, BD is vertical. Then sides AB and DC? Wait, AB is from (-1,1) to (2,4), DC is from (2,-4) to (-1,-1). Are AB and DC parallel? Let me check their slopes. AB slope is 1, DC slope is (-1 - (-4))/(-1 - 2) = (3)/(-3) = -1. Not parallel. So not a trapezoid. So maybe it's a kite? Let me check. A kite has two pairs of adjacent sides equal. Let me compute lengths.

AB: distance between A(-1,1) and B(2,4): sqrt[(2 - (-1))² + (4 - 1)²] = sqrt[9 + 9] = sqrt(18) = 3√2.

AD: distance between A(-1,1) and D(2,-4): sqrt[(2 - (-1))² + (-4 - 1)²] = sqrt[9 + 25] = sqrt(34).

BC: distance between B(2,4) and C(-1,-1): sqrt[(-1 - 2)^2 + (-1 - 4)^2] = sqrt[9 + 25] = sqrt(34).

CD: distance between C(-1,-1) and D(2,-4): sqrt[(2 - (-1))² + (-4 - (-1))²] = sqrt[9 + 9] = sqrt(18) = 3√2.

So sides AB and CD are equal (3√2), and sides AD and BC are equal (sqrt(34)). Therefore, it is a kite. Therefore, the area of a kite is (d1*d2)/2, where d1 and d2 are the lengths of the diagonals. Let me find the diagonals. The diagonals are AC and BD.

Diagonal AC: from A(-1,1) to C(-1,-1). This is vertical line x=-1, from y=1 to y=-1. Length is 2 units.

Diagonal BD: from B(2,4) to D(2,-4). This is vertical line x=2, from y=4 to y=-4. Length is 8 units.

Wait, so diagonals are AC (length 2) and BD (length 8). Therefore, area is (2 * 8)/2 = 8. Wait, but earlier shoelace formula gave 15? There's a contradiction here. Which one is correct?

Wait, if it's a kite with diagonals 2 and 8, area is 8. But shoelace formula gave 15. There must be a mistake in one of the methods. Let me check the diagonals again. Wait, in a kite, the diagonals are perpendicular? Wait, in a kite, one diagonal is the axis of symmetry. In this case, since the kite is symmetric over the x-axis, the diagonals should be AC and BD. Wait, AC is vertical line x=-1, BD is vertical line x=2. Wait, but they are both vertical, so they are parallel? That can't be. Wait, if both diagonals are vertical, then they are parallel, which would mean the kite is degenerate? But that can't be. Wait, something is wrong here. Wait, if the diagonals are AC and BD, but AC is from (-1,1) to (-1,-1), BD is from (2,4) to (2,-4). These are two vertical lines, parallel, so they don't intersect. Therefore, this is not a kite. Wait, but earlier calculation showed that AB = CD and AD = BC. So maybe it's a parallelogram? Wait, in a parallelogram, opposite sides are equal and parallel. But AB and CD are equal but not parallel (slopes 1 and -1). AD and BC are equal but not parallel (slopes -5/3 and 5/3). Therefore, not a parallelogram. Therefore, my previous conclusion that it's a kite is wrong? Wait, how? If AB = CD and AD = BC, then it's a parallelogram? Wait, no. In a parallelogram, opposite sides are equal and parallel. Here, AB and CD are equal but not parallel. AD and BC are equal but not parallel. Therefore, it's not a parallelogram. So maybe it's a different type of quadrilateral. Then why did I think it's a kite? Because two pairs of adjacent sides are equal? Wait, in a kite, two distinct pairs of adjacent sides are equal. In this case, AB = CD and AD = BC. But AB and AD are adjacent sides? Wait, AB is adjacent to AD? Yes. So AB = CD and AD = BC. But in a kite, the two pairs of adjacent sides are equal. However, in this case, AB and AD are adjacent sides, but AB ≠ AD. Wait, AB is 3√2, AD is sqrt(34). So they are not equal. Therefore, this is not a kite. Therefore, my mistake earlier. So the quadrilateral is not a kite. Therefore, the area calculation via shoelace formula is more reliable. But why did I get 15? Let me check again.

Wait, when I used the shoelace formula with the order A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A, I got total sum -30, absolute value 30, divided by 2 is 15. But if I think of the coordinates, the figure seems to have a larger area. Let me try to compute the area by breaking it into two triangles. For example, divide the quadrilateral into triangle ABC and triangle ACD? Wait, but need to make sure the triangles don't overlap. Alternatively, divide into triangle ABD and triangle BCD? Let me try.

First, triangle ABD: points A(-1,1), B(2,4), D(2,-4). Area can be calculated using determinant formula:

Area = 1/2 | (x_A(y_B - y_D) + x_B(y_D - y_A) + x_D(y_A - y_B)) |

Plugging in:

1/2 | (-1)(4 - (-4)) + 2*(-4 - 1) + 2*(1 - 4) | = 1/2 | (-1)(8) + 2*(-5) + 2*(-3) | = 1/2 | -8 -10 -6 | = 1/2 | -24 | = 12.

Then triangle BCD: points B(2,4), C(-1,-1), D(2,-4). Area:

1/2 | x_B(y_C - y_D) + x_C(y_D - y_B) + x_D(y_B - y_C) |.

Plugging in:

1/2 | 2*(-1 - (-4)) + (-1)*(-4 - 4) + 2*(4 - (-1)) | = 1/2 | 2*(3) + (-1)*(-8) + 2*(5) | = 1/2 | 6 + 8 + 10 | = 1/2 * 24 = 12.

Total area: 12 + 12 = 24. Wait, but this contradicts the shoelace formula result of 15. Which is correct? There must be an error here. Wait, if I split the quadrilateral into triangles ABD and BCD, but depending on how the quadrilateral is shaped, maybe these triangles overlap or something? Wait, if the quadrilateral is A(-1,1), B(2,4), D(2,-4), C(-1,-1), then splitting into ABD and BCD would cover the entire quadrilateral? Let me visualize. From A to B to D to C to A. If I split into ABD and BCD, then yes, they should cover the quadrilateral. But why different results? Let me check the area of triangle ABD again.

Alternatively, use coordinates for triangle ABD. Coordinates A(-1,1), B(2,4), D(2,-4). Using shoelace formula for triangle:

List points A(-1,1), B(2,4), D(2,-4), back to A(-1,1).

Compute terms:

(-1)(4) - 2(1) = -4 - 2 = -6

2*(-4) - 2*4 = -8 - 8 = -16

2*1 - (-1)*(-4) = 2 - 4 = -2

Sum: -6 -16 -2 = -24. Absolute value / 2: 12. Correct. Similarly for triangle BCD: points B(2,4), C(-1,-1), D(2,-4). Shoelace formula:

2*(-1) - (-1)*4 = -2 + 4 = 2

(-1)*(-4) - 2*(-1) = 4 + 2 = 6

2*4 - 2*(-4) = 8 + 8 = 16

Wait, no, let me do it properly. Shoelace formula for triangle BCD:

Coordinates B(2,4), C(-1,-1), D(2,-4), back to B(2,4).

First term: x_B * y_C - x_C * y_B = 2*(-1) - (-1)*4 = -2 + 4 = 2

Second term: x_C * y_D - x_D * y_C = (-1)*(-4) - 2*(-1) = 4 + 2 = 6

Third term: x_D * y_B - x_B * y_D = 2*4 - 2*(-4) = 8 + 8 = 16

Sum: 2 + 6 + 16 = 24. Absolute value / 2: 12. So total area 12 + 12 = 24. But earlier shoelace formula gave 15. There's a contradiction here. Which is correct?

Wait, maybe the order of the points in the shoelace formula was wrong. When I did the shoelace formula for the quadrilateral A, B, D, C, I got -30, which gave 15. But splitting into two triangles gives 24. Which is correct? Let me check with coordinates.

Alternatively, use vector cross product. The area of quadrilateral can be calculated as the sum of the areas of triangles ABD and BCD. If that gives 24, then that's the correct area. But why does the shoelace formula give 15? Because the order of the points was incorrect, leading to overlapping or incorrect traversal. Let me try again with the shoelace formula, making sure the points are ordered correctly.

Let me order the points in a clockwise or counter-clockwise order. Let me try A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A(-1,1). Let me plot these points:

Starting at A(-1,1), moving to B(2,4): upper right.

Then to D(2,-4): straight down along x=2 to (2,-4).

Then to C(-1,-1): moving left and up to (-1,-1).

Then back to A(-1,1): up along x=-1.

This path seems to create a quadrilateral that is a non-intersecting polygon. Let me apply shoelace formula again carefully.

List the coordinates:

A(-1,1)

B(2,4)

D(2,-4)

C(-1,-1)

A(-1,1)

Compute shoelace sum:

First, multiply xi * yi+1:

(-1)*4 + 2*(-4) + 2*(-1) + (-1)*1 = (-4) + (-8) + (-2) + (-1) = -15

Then multiply yi * xi+1:

1*2 + 4*2 + (-4)*(-1) + (-1)*(-1) = 2 + 8 + 4 + 1 = 15

Subtract the two: | -15 - 15 | = | -30 | = 30. Divide by 2: 15. So area is 15. But splitting into triangles gives 24. There's a contradiction. Which is correct?

Wait, maybe the problem is that when I split into triangles ABD and BCD, I'm actually calculating the area of a different quadrilateral? Wait, if the quadrilateral is A, B, D, C, then splitting into ABD and BCD should be correct. But why discrepancy?

Wait, let me check coordinates again. If I plot the points:

A(-1,1), B(2,4), D(2,-4), C(-1,-1). Connecting A to B to D to C to A. This creates a quadrilateral that is a bowtie? Wait, no. Wait, connecting A(-1,1) to B(2,4) to D(2,-4) to C(-1,-1) to A. Let me imagine this. From A(-1,1) to B(2,4): upper right. Then to D(2,-4): straight down. Then to C(-1,-1): left and up. Then back to A(-1,1): up. This seems to create a quadrilateral that is a trapezoid? Or maybe not. Wait, if I imagine this, the sides AB and DC are not parallel, but the sides AD and BC are not parallel. However, when I split into triangles ABD and BCD, I get 24, but shoelace formula gives 15. There must be an error in one of the methods.

Wait, let me check the shoelace formula again. Maybe I made a mistake in the calculation. Let me redo it step by step.

Shoelace formula for quadrilateral A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A(-1,1):

List the coordinates:

x: -1, 2, 2, -1, -1

y: 1, 4, -4, -1, 1

Compute sum of xi * yi+1:

(-1)*4 + 2*(-4) + 2*(-1) + (-1)*1 = (-4) + (-8) + (-2) + (-1) = -15

Compute sum of yi * xi+1:

1*2 + 4*2 + (-4)*(-1) + (-1)*(-1) = 2 + 8 + 4 + 1 = 15

Subtract: | -15 - 15 | = 30. Area = 30 / 2 = 15. So shoelace formula says 15.

But splitting into triangles gives 24. Where is the mistake?

Wait, maybe the triangles overlap? If I split the quadrilateral into ABD and BCD, maybe they overlap? Let me check. The triangle ABD is formed by points A, B, D. The triangle BCD is formed by points B, C, D. These two triangles share the side BD. So together they form the quadrilateral A, B, D, C. So they should not overlap. Therefore, their areas should add up. But why discrepancy?

Wait, let me recalculate the area of triangle ABD. Points A(-1,1), B(2,4), D(2,-4). Using shoelace formula:

Coordinates:

A(-1,1), B(2,4), D(2,-4), back to A(-1,1).

Compute sum of xi*yi+1:

(-1)*4 + 2*(-4) + 2*1 = -4 -8 + 2 = -10

Sum of yi*xi+1:

1*2 + 4*2 + (-4)*(-1) = 2 + 8 + 4 = 14

Subtract: |-10 - 14| = 24. Area = 24 / 2 = 12. Correct.

Triangle BCD: points B(2,4), C(-1,-1), D(2,-4). Shoelace formula:

Coordinates:

B(2,4), C(-1,-1), D(2,-4), back to B(2,4).

Sum of xi*yi+1:

2*(-1) + (-1)*(-4) + 2*4 = -2 + 4 + 8 = 10

Sum of yi*xi+1:

4*(-1) + (-1)*2 + (-4)*2 = -4 -2 -8 = -14

Subtract: |10 - (-14)| = |24|, area = 24 / 2 = 12. So total area 12 + 12 = 24. But shoelace formula for quadrilateral gives 15. This is a contradiction. There must be a mistake in the order of the points in the shoelace formula for the quadrilateral. Maybe the order is not correct, leading to the shoelace formula calculating the area of a different polygon?

Wait, if I take the order A, B, D, C, the shoelace formula gives 15, but splitting into triangles gives 24. Let me try another order. What if I take the order A, B, C, D? Let me try that.

Order: A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A(-1,1).

Compute shoelace formula:

Sum of xi*yi+1:

(-1)*4 + 2*(-1) + (-1)*(-4) + 2*1 = -4 -2 + 4 + 2 = 0

Sum of yi*xi+1:

1*2 + 4*(-1) + (-1)*2 + (-4)*(-1) = 2 -4 -2 + 4 = 0

Subtract: |0 - 0| = 0. Area 0. Which is wrong. So this order is incorrect.

Another order: A, C, B, D.

Order: A(-1,1), C(-1,-1), B(2,4), D(2,-4), back to A.

Compute shoelace:

Sum xi*yi+1:

(-1)*(-1) + (-1)*4 + 2*(-4) + 2*1 = 1 -4 -8 + 2 = -9

Sum yi*xi+1:

1*(-1) + (-1)*2 + 4*2 + (-4)*(-1) = -1 -2 + 8 + 4 = 9

Subtract: |-9 - 9| = 18. Area 9. Still not matching.

Another order: A, D, B, C.

Order: A(-1,1), D(2,-4), B(2,4), C(-1,-1), back to A.

Compute:

Sum xi*yi+1:

(-1)*(-4) + 2*4 + 2*(-1) + (-1)*1 = 4 + 8 - 2 -1 = 9

Sum yi*xi+1:

1*2 + (-4)*2 + 4*(-1) + (-1)*(-1) = 2 -8 -4 + 1 = -9

Subtract: |9 - (-9)| = 18. Area 9. Still not matching.

Wait, so why does splitting into triangles give 24, but shoelace formula with different orders gives 15 or 9 or 0? It seems that the correct area is 24, and the shoelace formula was giving wrong results because of incorrect ordering. How to determine the correct order?

Let me think of the quadrilateral as a non-intersecting polygon. If I traverse the points in the order A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A, does this create a non-intersecting polygon? Let me imagine:

Start at A(-1,1), go to B(2,4), then to D(2,-4), then to C(-1,-1), then back to A. This path would cross over itself? Let me check. From A to B to D: that's a line from A to B to D. Then from D to C to A: from D(2,-4) to C(-1,-1) to A(-1,1). This seems to create a quadrilateral that is a non-intersecting polygon. But why does the shoelace formula give 15? Maybe I made a mistake in the shoelace calculation. Let me try once more with this order.

Order: A(-1,1), B(2,4), D(2,-4), C(-1,-1), A(-1,1).

Compute shoelace formula step by step:

List the coordinates:

x: -1, 2, 2, -1, -1

y: 1, 4, -4, -1, 1

Compute sum of xi * yi+1:

(-1)*4 + 2*(-4) + 2*(-1) + (-1)*1 = (-4) + (-8) + (-2) + (-1) = -15

Compute sum of yi * xi+1:

1*2 + 4*2 + (-4)*(-1) + (-1)*(-1) = 2 + 8 + 4 + 1 = 15

Subtract: |-15 - 15| = 30. Area = 30 / 2 = 15.

But splitting into triangles gives 24. This is very confusing. Let me try to calculate the area using coordinates with vectors or integration.

Alternatively, think of the quadrilateral as a polygon with vertices at (-1,1), (2,4), (2,-4), (-1,-1). Let me use the shoelace formula with a different order that ensures the polygon is traversed correctly. Let me try the order A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A. Wait, earlier this gave zero. But maybe I did it wrong. Let me recalculate:

Order: A(-1,1), B(2,4), C(-1,-1), D(2,-4), A(-1,1).

Compute xi*yi+1:

(-1)*4 + 2*(-1) + (-1)*(-4) + 2*1 = -4 -2 + 4 + 2 = 0

Compute yi*xi+1:

1*2 + 4*(-1) + (-1)*2 + (-4)*(-1) = 2 -4 -2 + 4 = 0

Subtract: 0 - 0 = 0. Area 0. This is wrong. So this order is incorrect.

Wait, maybe the correct order is A(-1,1), B(2,4), D(2,-4), C(-1,-1), A(-1,1). But why does this give 15? Let me try to visualize the area. If I imagine the quadrilateral, it's like a bowtie? No, because it's non-intersecting. Wait, if I connect A to B to D to C to A, it should be a quadrilateral. Let me calculate the area using coordinates with vectors.

Alternatively, use the shoelace formula with the correct order. Maybe I need to ensure that the points are ordered either clockwise or counter-clockwise without crossing.

Let me try ordering the points as A(-1,1), B(2,4), D(2,-4), C(-1,-1), back to A. Let me plot this:

Start at A(-1,1), go to B(2,4): upper right.

From B(2,4) to D(2,-4): straight down along x=2.

From D(2,-4) to C(-1,-1): left and up.

From C(-1,-1) back to A(-1,1): up along x=-1.

This should form a quadrilateral that is a trapezoid? Wait, no. Wait, if you imagine this, the sides AB and DC are not parallel, but the sides AD and BC are not parallel. However, the area calculated by shoelace is 15, while splitting into triangles gives 24. This is very confusing. Maybe I need to check with coordinates.

Let me calculate the area using the shoelace formula with a different approach. Let me use the coordinates and calculate the area step by step.

The shoelace formula is sensitive to the order of the points. If the points are ordered incorrectly, it can give wrong results. Let me try to order the points in a counter-clockwise manner.

Let me list the points:

A(-1,1), B(2,4), D(2,-4), C(-1,-1).

To order them counter-clockwise, let's imagine the positions:

A is on the left upper, B is on the right upper, D is on the right lower, C is on the left lower.

A counter-clockwise order would be A, B, D, C. Let me check:

Starting at A(-1,1), moving to B(2,4) (upper right), then to D(2,-4) (down), then to C(-1,-1) (left and up), then back to A. This seems counter-clockwise? Or is it clockwise? It's hard to tell without a diagram. Let me use the shoelace formula with this order and see.

As before, it gives 15. But splitting into triangles gives 24. There must be a mistake in one of the methods. Let me try to calculate the area using coordinates with vectors.

The area can also be calculated using the vector cross product. For a polygon, the area is 1/2 |sum_{i=1 to n} (xi*yi+1 - xi+1*yi)|. So if I trust the shoelace formula, it's 15. But why does splitting into triangles give 24? Let me check the coordinates again.

Wait, if I take the quadrilateral A(-1,1), B(2,4), D(2,-4), C(-1,-1), and calculate the area using the shoelace formula, it's 15. But if I calculate the area of the triangle ABD (12) and BCD (12), it's 24. This suggests that the shoelace formula is giving the area of a different quadrilateral. Perhaps the order of the points in the shoelace formula is creating a self-intersecting polygon, which results in subtracting areas?

Wait, if the order A, B, D, C creates a self-intersecting polygon, then the shoelace formula would calculate the net area, which could be less than the actual area. Let me check if the polygon is self-intersecting. From A(-1,1) to B(2,4) to D(2,-4) to C(-1,-1) to A(-1,1). Does this cross over? Let me see: the line from B(2,4) to D(2,-4) is vertical. The line from D(2,-4) to C(-1,-1) goes from (2,-4) to (-1,-1). The line from C(-1,-1) to A(-1,1) is vertical. The line from A(-1,1) to B(2,4) is diagonal. I don't see any intersections. So it's a simple quadrilateral. Therefore, the shoelace formula should give the correct area. But why discrepancy with the triangle areas?

Wait, let me calculate the area of the quadrilateral using another method. Let me use coordinates to integrate or use the formula for the area of a trapezoid. Wait, but it's not a trapezoid. Alternatively, use coordinates to find the area.

Let me consider the coordinates of the quadrilateral. It has vertices at (-1,1), (2,4), (2,-4), (-1,-1). Let me plot these points on a coordinate system.

Point A(-1,1) is on the left side, upper.

Point B(2,4) is on the right side, upper.

Point D(2,-4) is on the right side, lower.

Point C(-1,-1) is on the left side, lower.

Connecting these points in order A-B-D-C-A. This creates a quadrilateral that is symmetric about the x-axis? Wait, no. Wait, point A(-1,1) and C(-1,-1) are symmetric about the x-axis. Point B(2,4) and D(2,-4) are symmetric about the x-axis. So the entire quadrilateral is symmetric about the x-axis. Therefore, the area should be twice the area of the upper half. Let me calculate the area of the upper half.

The upper half consists of the points A(-1,1), B(2,4), and the x-axis. Wait, no. The upper half of the quadrilateral would be the area between A(-1,1), B(2,4), and the line connecting B(2,4) to the x-axis? No, it's better to think of it as the area above the x-axis. Since the quadrilateral is symmetric, the area above the x-axis is equal to the area below. Let me calculate the area above the x-axis.

The upper part of the quadrilateral is the polygon formed by A(-1,1), B(2,4), and the intersection of the line BD with the x-axis? Wait, no. Wait, the quadrilateral is A-B-D-C-A. The part above the x-axis is the triangle A-B-D? No, because D is below the x-axis. Wait, this is getting too confusing. Let me use the symmetry. Since the quadrilateral is symmetric about the x-axis, the area should be twice the area of the part above the x-axis. The part above the x-axis is the area bounded by A(-1,1), B(2,4), and the x-axis? No, it's bounded by A(-1,1), B(2,4), and the line from B(2,4) to D(2,-4) which crosses the x-axis at (2,0). Wait, this is getting too complex. Let me instead calculate the area of the upper half by integrating or using coordinates.

Alternatively, since the quadrilateral is symmetric, let me calculate the area of the upper half, which is the area between A(-1,1), B(2,4), and the x-axis. Wait, but how? The upper half of the quadrilateral would be the area bounded by A(-1,1), B(2,4), and the line from B(2,4) to the x-axis? No, I need to define it properly.

Alternatively, since the quadrilateral is symmetric, the area above the x-axis is the area of the triangle formed by A(-1,1), B(2,4), and the projection of B on the x-axis, which is (2,0). Similarly, the area below the x-axis is the same. But this might not be accurate.

Let me calculate the area above the x-axis. It would be the area of the polygon A(-1,1), B(2,4), (2,0), and back to A(-1,1)? No, not sure. This is getting too complicated. Let me go back to the shoelace formula. If the shoelace formula with the correct order gives 15, and splitting into triangles gives 24, there must be a mistake in one of the calculations. Let me check the triangle areas again.

For triangle ABD: points A(-1,1), B(2,4), D(2,-4). Using the formula for area of a triangle with coordinates:

Area = 1/2 | (x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) |

Plugging in:

1/2 | (-1)(4 - (-4)) + 2(-4 - 1) + 2(1 - 4) | = 1/2 | (-1)(8) + 2(-5) + 2(-3) | = 1/2 | -8 -10 -6 | = 1/2 | -24 | = 12. Correct.

For triangle BCD: points B(2,4), C(-1,-1), D(2,-4). Using the same formula:

1/2 | 2(-1 - (-4)) + (-1)(-4 - 4) + 2(4 - (-1)) | = 1/2 | 2(3) + (-1)(-8) + 2(5) | = 1/2 | 6 + 8 + 10 | = 1/2 * 24 = 12. Correct.

So total area 24. But why does the shoelace formula give 15? It must be that the order of the points in the shoelace formula is incorrect, leading to the calculation of a different area. Let me try to find the correct order.

Let me try ordering the points as A(-1,1), B(2,4), C(-1,-1), D(2,-4), back to A. Wait, this gave zero. Let me try another order: A(-1,1), D(2,-4), C(-1,-1), B(2,4), back to A. Let's apply shoelace formula.

Coordinates: A(-1,1), D(2,-4), C(-1,-1), B(2,4), A(-1,1).

Compute xi*yi+1:

(-1)*(-4) + 2*(-1) + (-1)*4 + 2*1 = 4 - 2 - 4 + 2 = 0

Compute yi*xi+1:

1*2 + (-4)*(-1) + (-1)*2 + 4*(-1) = 2 + 4 - 2 - 4 = 0

Subtract: 0 - 0 = 0. Area 0. Still wrong.

Wait, maybe the correct order is A(-1,1), B(2,4), D(2,-4), C(-1,-1), A(-1,1). Let me try to visualize this as a non-intersecting polygon. If I go from A to B to D to C to A, it should be a quadrilateral. Let me calculate the area using the shoelace formula with this order, but this time, I'll use the formula step by step.

List of points:

1. A(-1,1)
2. B(2,4)
3. D(2,-4)
4. C(-1,-1)
5. A(-1,1)

Compute the shoelace sum:

Sum1 = (x1*y2 + x2*y3 + x3*y4 + x4*y5)

= (-1)*4 + 2*(-4) + 2*(-1) + (-1)*1 = -4 -8 -2 -1 = -15

Sum2 = (y1*x2 + y2*x3 + y3*x4 + y4*x5)

= 1*2 + 4*2 + (-4)*(-1) + (-1)*(-1) = 2 + 8 + 4 + 1 = 15

Area = 1/2 |Sum1 - Sum2| = 1/2 |-15 - 15| = 1/2 * 30 = 15.

But why is this different from the triangle method? It must be that the shoelace formula is calculating the area of a different polygon. Wait, if I connect A to B to D to C to A, it's possible that this creates a self-intersecting polygon, but I don't see how. Let me check if the lines cross. The line from B(2,4) to D(2,-4) is vertical. The line from D(2,-4) to C(-1,-1) is from (2,-4) to (-1,-1). The line from C(-1,-1) to A(-1,1) is vertical. The line from A(-1,1) to B(2,4) is diagonal. No intersections. So it's a simple quadrilateral. Therefore, the area should be 15. But why does splitting into triangles give 24? This is very confusing.

Wait, let me calculate the area using the shoelace formula with the correct order, and also calculate the area using the triangle method, but with the correct triangles. Maybe I chose the wrong triangles. Let me think: if the quadrilateral is A-B-D-C, then the correct way to split it into triangles is A-B-D and A-D-C? Or A-B-C and B-C-D? Wait, no. If the quadrilateral is A-B-D-C, then the diagonals are A-D and B-C. So splitting into triangles A-B-D and A-D-C. Let me try that.

Triangle A-B-D: area 12.

Triangle A-D-C: points A(-1,1), D(2,-4), C(-1,-1). Let me calculate this area.

Using the formula:

1/2 | x_A(y_D - y_C) + x_D(y_C - y_A) + x_C(y_A - y_D) |

= 1/2 | (-1)(-4 - (-1)) + 2(-1 - 1) + (-1)(1 - (-4)) |

= 1/2 | (-1)(-3) + 2(-2) + (-1)(5) |

= 1/2 | 3 - 4 - 5 | = 1/2 | -6 | = 3.

So total area would be 12 + 3 = 15. Ah! Now it matches the shoelace formula. So earlier, I was splitting into triangles ABD and BCD, which were not part of the quadrilateral A-B-D-C. Instead, the correct split is ABD and ADC, giving 12 + 3 = 15. So my mistake was in choosing the wrong triangles. Earlier, I split into ABD and BCD, but BCD is not part of the quadrilateral A-B-D-C. The correct split is ABD and ADC, giving 15. Therefore, the correct area is 15.

So the area of the quadrilateral is 15. But let me confirm with the shoelace formula. Yes, it gave 15. So the correct area is 15. Therefore, the answer is 15. But let me check with the Asymptote code. The Asymptote code draws points A(-1,1), B(2,4), C(-1,-1), D(2,-4), and the area is 15. But the problem says "the area of the quadrilateral formed by the four points of tangency." So the answer is 15? But let me check with the original problem.

Wait, the original problem says the circle x² + y² = 2 and the parabola y² = 8x have two common tangents, forming four points of tangency. Find the area of the quadrilateral formed by the four points of tangency.

But according to our calculation, the area is 15. But let me check if the points A(-1,1), B(2,4), C(-1,-1), D(2,-4) are indeed the points of tangency. Earlier, we checked that the line AB is tangent to both the circle and the parabola, and similarly for CD. So these are the points of tangency. Therefore, the quadrilateral formed by these four points has area 15. But let me check if this is correct.

Wait, but the circle has radius sqrt(2) ≈ 1.414, and the points A(-1,1) and C(-1,-1) are on the circle. Let me check the distance from the origin to A: sqrt((-1)^2 + 1^2) = sqrt(2), correct. Similarly for C. Points B(2,4) and D(2,-4) are on the parabola y² = 8x: 4² = 16, 8*2 = 16, correct. So these are indeed the points of tangency. Therefore, the area is 15. But let me check with another method.

Alternatively, since the quadrilateral is symmetric about the x-axis, we can calculate the area of the upper half and double it. The upper half is the area between A(-1,1), B(2,4), and the x-axis. Wait, but how? Let me calculate the area of the upper part as the area of the triangle ABD, but no. Wait, the upper part is the area bounded by A(-1,1), B(2,4), and the line from B(2,4) to the x-axis. Let me calculate this.

The line from B(2,4) to the x-axis is the vertical line x=2 from (2,4) to (2,0). The area of the upper part would be the area of the trapezoid formed by A(-1,1), B(2,4), (2,0), and the projection of A on the x-axis, which is (-1,0). But this is getting complex. Let me calculate it.

The area of the upper part can be calculated as the area of the trapezoid with vertices at A(-1,1), B(2,4), (2,0), and (-1,0). The area of this trapezoid is the average of the two bases times the height. The bases are the lengths from (-1,1) to (-1,0) which is 1, and from (2,4) to (2,0) which is 4. The height is the horizontal distance between x=-1 and x=2, which is 3. Wait, no. The formula for the area of a trapezoid is (base1 + base2)/2 * height, where height is the distance between the two bases. In this case, the two bases are the vertical segments at x=-1 and x=2, with lengths 1 and 4. The distance between these bases is the horizontal distance between x=-1 and x=2, which is 3. So area is (1 + 4)/2 * 3 = 5/2 * 3 = 15/2. Then the total area would be twice this, 15. Which matches the previous result. Therefore, the area is indeed 15.

Therefore, the answer is 15. But let me check if the problem might expect a different answer. Wait, the Asymptote code in the problem has points A(-1,1), B(2,4), C(-1,-1), D(2,-4), and draws lines connecting A to B, C to D, and A to C, B to D. The area of the quadrilateral formed by these points is 15. Therefore, the answer should be 15. But let me check if there is a calculation error.

Wait, the trapezoid area calculation gave 15/2 for the upper half, so total area 15. Yes. And the shoelace formula and triangle split (correctly) also gave 15. Therefore, the correct area is 15. So the answer is 15, which in boxed form is \boxed{15}.

**Final Answer**
\boxed{15}

To find the area of the quadrilateral formed by the four points of tangency between the circle $ x^2 + y^2 = 2 $ and the parabola $ y^2 = 8x $, we begin by identifying the points of tangency.

---

### Step 1: Identify the Points of Tangency

We are told that the circle and the parabola have **two common tangents**, each of which touches both the circle and the parabola at one point. Thus, there are **four points of tangency** in total: two on the circle and two on the parabola.

From the Asymptote code and the problem's context, the points of tangency are:

- $ A = (-1, 1) $ — on the circle
- $ B = (2, 4) $ — on the parabola
- $ C = (-1, -1) $ — on the circle
- $ D = (2, -4) $ — on the parabola

These points form a quadrilateral $ ABCD $, which is symmetric about the x-axis.

---

### Step 2: Use the Shoelace Formula to Compute the Area

We apply the **shoelace formula** to compute the area of quadrilateral $ ABCD $, using the coordinates in the order $ A \to B \to D \to C \to A $:

$$
\text{Area} = \frac{1}{2} \left| \sum (x_i y_{i+1} - x_{i+1} y_i) \right|
$$

List of points in order:
- $ A = (-1, 1) $
- $ B = (2, 4) $
- $ D = (2, -4) $
- $ C = (-1, -1) $
- Back to $ A = (-1, 1) $

Compute the terms:

- $ x_A y_B = (-1)(4) = -4 $
- $ x_B y_D = (2)(-4) = -8 $
- $ x_D y_C = (2)(-1) = -2 $
- $ x_C y_A = (-1)(1) = -1 $

Sum of $ x_i y_{i+1} $:
$$
-4 - 8 - 2 - 1 = -15
$$

Now compute the other sum:

- $ y_A x_B = (1)(2) = 2 $
- $ y_B x_D = (4)(2) = 8 $
- $ y_D x_C = (-4)(-1) = 4 $
- $ y_C x_A = (-1)(-1) = 1 $

Sum of $ y_i x_{i+1} $:
$$
2 + 8 + 4 + 1 = 15
$$

Now compute the area:

$$
\text{Area} = \frac{1}{2} | -15 - 15 | = \frac{1}{2} \cdot 30 = 15
$$

---

### Step 3: Confirm with Triangle Decomposition

To verify, we can split the quadrilateral into two triangles:

- Triangle $ ABD $: Points $ A(-1,1), B(2,4), D(2,-4) $
- Triangle $ ACD $: Points $ A(-1,1), C(-1,-1), D(2,-4) $

Using the shoelace formula for each triangle:

- Area of $ ABD $: $ \frac{1}{2} | -24 | = 12 $
- Area of $ ACD $: $ \frac{1}{2} | -6 | = 3 $

Total area: $ 12 + 3 = 15 $

---

### Final Answer

$$
\boxed{15}
$$

Solution:
Let the equation of a tangent line be $y = mx + b.$

Substituting into the equation $x^2 + y^2 = 2,$ we get
\[x^2 + (mx + b)^2 = 2.\]Then $(m^2 + 1) x^2 + 2bmx + (b^2 - 2) = 0.$  Since we have a tangent, this quadratic has a double root, meaning that its discriminant is 0.  This gives us
\[(2bm)^2 - 4(m^2 + 1)(b^2 - 2) = 0,\]which simplifies to $b^2 = 2m^2 + 2.$

Solving for $x$ in $y = mx + b,$ we get $x = \frac{y - b}{m}.$  Substituting into $y^2 = 8x,$ we get
\[y^2 = \frac{8y - 8b}{m},\]so $my^2 - 8y + 8b = 0.$  Again, the discriminant of this quadratic will also be 0, so
\[64 - 4(m)(8b) = 0.\]Hence, $bm = 2.$

Then $b = \frac{2}{m}.$  Substituting into $b^2 = 2m^2 + 2,$ we get
\[\frac{4}{m^2} = 2m^2 + 2.\]Then $4 = 2m^4 + 2m^2,$ so $m^4 + m^2 - 2 = 0.$  This factors as $(m^2 - 1)(m^2 + 2) = 0.$  Hence, $m^2 = 1,$ so $m = \pm 1.$

If $m = 1,$ then $b = 2.$  If $m = -1,$ then $b = -2.$  Thus, the two tangents are $y = x + 2$ and $y = -x - 2.$

[asy]
unitsize(0.8 cm);

real upperparab (real x) {
  return (sqrt(8*x));
}

real lowerparab (real x) {
  return (-sqrt(8*x));
}

pair A, B, C, D;

A = (-1,1);
B = (2,4);
C = (-1,-1);
D = (2,-4);

draw(graph(upperparab,0,3));
draw(graph(lowerparab,0,3));
draw(Circle((0,0),sqrt(2)));
draw(interp(A,B,-0.2)--interp(A,B,1.2));
draw(interp(C,D,-0.2)--interp(C,D,1.2));
draw(A--C);
draw(B--D);

label("$y = x + 2$", interp(A,B,1.3), NE);
label("$y = -x - 2$", interp(C,D,1.3), SE);

dot(A);
dot(B);
dot(C);
dot(D);
[/asy]

We look at the tangent $y = x + 2.$  Substituting into $x^2 + y^2 = 2,$ we get
\[x^2 + (x + 2)^2 = 2.\]This simplifies to $x^2 + 2x + 1 = (x + 1)^2 = 0,$ so $x = -1.$  Hence, the tangent point on the circle is $(-1,1).$

We have that $x = y - 2.$  Substituting into $y^2 = 8x,$ we get
\[y^2 = 8(y - 2).\]This simplifies to $(y - 4)^2 = 0,$ so $y = 4.$  Hence, the tangent point on the parabola is $(2,4).$

By symmetry, the other two tangent points are $(-1,-1)$ and $(2,-4).$

The quadrilateral in question is a trapezoid with bases 2 and 8, and height 3, so its area is $\frac{2 + 8}{2} \cdot 3 = \boxed{15}.$