Explanation:
Okay, so I need to find the value of f(3) given that f(-3) = 2 for the function f(x) = ax⁴ - bx² + x + 5. Hmm, let me think about how to approach this.

First, let me write down the function again to make sure I have it right: f(x) = a x⁴ - b x² + x + 5. So it's a quartic function with coefficients a and b. The problem gives me f(-3) = 2, and I need to find f(3).

Since the function has both even and odd powers of x, maybe there's some symmetry I can exploit here. Let me recall that for functions involving even and odd powers, sometimes substituting -x can lead to simplifications. Let me try that.

Let me compute f(-x). If I replace x with -x in the function, I get:

f(-x) = a(-x)⁴ - b(-x)² + (-x) + 5.

Calculating each term:

(-x)⁴ is x⁴ because any even power of a negative number is positive. Similarly, (-x)² is x². So substituting back:

f(-x) = a x⁴ - b x² - x + 5.

Now, let me compare this with the original function f(x) = a x⁴ - b x² + x + 5.

So, if I subtract f(-x) from f(x), let's see what happens:

f(x) - f(-x) = [a x⁴ - b x² + x + 5] - [a x⁴ - b x² - x + 5].

Let me compute term by term:

a x⁴ - a x⁴ = 0.

- b x² - (-b x²) = -b x² + b x² = 0.

Then, x - (-x) = x + x = 2x.

And 5 - 5 = 0.

So overall, f(x) - f(-x) = 2x.

That's interesting. So this tells me that f(x) - f(-x) = 2x.

Alternatively, maybe I can express f(x) + f(-x) as well? Let me check:

f(x) + f(-x) = [a x⁴ - b x² + x + 5] + [a x⁴ - b x² - x + 5] = 2a x⁴ - 2b x² + 10.

So that's another relation.

But how does this help me? Well, the problem gives me f(-3) = 2, and I need to find f(3). Let me denote f(3) as some value, say, y. Then, maybe I can set up equations using the relations I found.

Let me write down f(-3) and f(3) using the expressions above.

First, let's compute f(-3):

f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5 = a*(81) - b*(9) - 3 + 5 = 81a - 9b + 2.

And according to the problem, this equals 2. So:

81a - 9b + 2 = 2.

Subtracting 2 from both sides:

81a - 9b = 0.

Simplify this equation. Let me factor out 9:

9*(9a - b) = 0 => 9a - b = 0 => b = 9a.

Okay, so that's one equation relating a and b.

Now, let me compute f(3):

f(3) = a*(3)^4 - b*(3)^2 + 3 + 5 = 81a - 9b + 8.

Wait, so f(3) = 81a - 9b + 8.

But from earlier, we found that 81a - 9b = 0. Therefore, substituting that into f(3):

f(3) = 0 + 8 = 8.

Wait, so is it that straightforward? Let me check again.

Given that f(-3) = 81a - 9b - 3 + 5 = 81a - 9b + 2 = 2. Therefore, 81a - 9b = 0. Then f(3) is 81a - 9b + 3 + 5 = 81a - 9b + 8. Since 81a - 9b is zero, f(3) is 0 + 8 = 8.

So the answer is 8? That seems too straightforward. Let me verify with another approach to make sure I didn't make a mistake.

Alternatively, using the relation f(x) - f(-x) = 2x. Let me plug in x = 3. Then:

f(3) - f(-3) = 2*3 = 6.

We know that f(-3) = 2, so:

f(3) - 2 = 6 => f(3) = 6 + 2 = 8.

Yes! That's the same result. So this confirms that f(3) is 8.

Therefore, regardless of the values of a and b, as long as they satisfy the condition given by f(-3) = 2, the value of f(3) is 8.

Let me check if there's another way to think about this. For example, considering the function's even and odd parts.

The function f(x) can be split into even and odd components. The even part would be the terms with even powers of x, and the odd part would be the terms with odd powers.

Looking at f(x) = ax⁴ - bx² + x + 5.

The even part is ax⁴ - bx² + 5, and the odd part is x.

So, f(x) = [ax⁴ - bx² + 5] + x.

Now, if we consider the even function E(x) = ax⁴ - bx² + 5 and the odd function O(x) = x.

Then, f(-x) = E(-x) + O(-x) = E(x) - O(x).

Therefore, f(-x) = E(x) - O(x).

So, f(x) + f(-x) = 2E(x), and f(x) - f(-x) = 2O(x).

Which is consistent with what I found earlier.

So, since O(x) = x, then f(x) - f(-x) = 2x.

Therefore, if we take x = 3, then f(3) - f(-3) = 6. Since f(-3) = 2, then f(3) = 6 + 2 = 8.

Same result again.

Therefore, regardless of the values of a and b, as long as they satisfy the condition given by f(-3) = 2, the value of f(3) is 8.

Wait, but let me check if there's any possibility that the equations could have more constraints. For example, if I had more information, but here we only have one equation (from f(-3) = 2) and two variables (a and b). However, when calculating f(3), the terms involving a and b cancel out, so we don't need to know their exact values.

Let me verify with actual numbers. Suppose I choose a value for a, then compute b, and check f(3).

For example, let me pick a = 1. Then from b = 9a, we have b = 9.

So then f(x) = x⁴ - 9x² + x + 5.

Compute f(-3):

(-3)^4 - 9*(-3)^2 + (-3) + 5 = 81 - 9*9 - 3 + 5 = 81 - 81 - 3 + 5 = 0 - 3 + 5 = 2.

Which matches the given condition. Now compute f(3):

3^4 - 9*(3)^2 + 3 + 5 = 81 - 81 + 3 + 5 = 0 + 8 = 8.

Perfect, that works.

Let me try another value for a. Let me choose a = 0. Then b = 9*0 = 0.

So f(x) = 0x⁴ - 0x² + x + 5 = x + 5.

Then f(-3) = -3 + 5 = 2, which is correct. Then f(3) = 3 + 5 = 8.

Same result.

Another example: let me take a = 2, so b = 18.

Then f(x) = 2x⁴ - 18x² + x + 5.

Compute f(-3):

2*(81) - 18*(9) + (-3) + 5 = 162 - 162 - 3 + 5 = 0 - 3 + 5 = 2.

Good. Then f(3) = 2*(81) - 18*(9) + 3 + 5 = 162 - 162 + 3 + 5 = 0 + 8 = 8.

Same answer again.

Therefore, regardless of the values of a and b, as long as they satisfy the condition from f(-3) = 2, f(3) is always 8.

Therefore, the answer is 8.

But just to make sure I didn't overlook anything, let me check if there's another way someone might approach this problem and perhaps make a mistake.

For example, someone might think that since f(-3) = 2, then f(3) is something else, maybe related to the coefficients. But since the function has both even and odd terms, the even terms will be the same for x and -x, while the odd terms will have opposite signs.

So, if I write f(3) = even_part(3) + odd_part(3), and f(-3) = even_part(-3) + odd_part(-3) = even_part(3) - odd_part(3).

Therefore, if I add f(3) and f(-3), I get 2*even_part(3). If I subtract f(-3) from f(3), I get 2*odd_part(3).

In this case, the odd_part(3) is just the coefficient of x times 3, which is 1*3 = 3. Therefore, f(3) - f(-3) = 2*3 = 6. Therefore, f(3) = f(-3) + 6 = 2 + 6 = 8.

This is another way to see it, using the even and odd parts.

So, all methods lead to the same conclusion. Therefore, I can be confident that the answer is 8.

I think that's thorough enough. I considered multiple approaches, checked with examples, and verified the relationships between f(x) and f(-x). All roads lead to Rome here.

**Final Answer**
The value of $ f(3) $ is \boxed{8}.

To determine the value of $ f(3) $, we are given the function:

$$
f(x) = ax^4 - bx^2 + x + 5
$$

and the condition:

$$
f(-3) = 2
$$

---

### Step 1: Use the given condition to find a relationship between $ a $ and $ b $

We substitute $ x = -3 $ into the function:

$$
f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5 = 81a - 9b - 3 + 5 = 81a - 9b + 2
$$

We are told that $ f(-3) = 2 $, so:

$$
81a - 9b + 2 = 2
$$

Subtracting 2 from both sides:

$$
81a - 9b = 0
$$

Divide both sides by 9:

$$
9a - b = 0 \quad \Rightarrow \quad b = 9a
$$

---

### Step 2: Use the relationship to find $ f(3) $

Now compute $ f(3) $:

$$
f(3) = a(3)^4 - b(3)^2 + 3 + 5 = 81a - 9b + 8
$$

Substitute $ b = 9a $:

$$
f(3) = 81a - 9(9a) + 8 = 81a - 81a + 8 = 8
$$

---

### Step 3: Confirm using symmetry

We can also observe that the function has both even and odd components:

- Even part: $ ax^4 - bx^2 + 5 $
- Odd part: $ x $

So, the function can be written as:

$$
f(x) = \text{even}(x) + \text{odd}(x)
$$

Then:

$$
f(-x) = \text{even}(x) - \text{odd}(x)
$$

So:

$$
f(x) - f(-x) = 2 \cdot \text{odd}(x) = 2x
$$

This gives us:

$$
f(3) - f(-3) = 2 \cdot 3 = 6
$$

Since $ f(-3) = 2 $, we find:

$$
f(3) = 2 + 6 = 8
$$

---

### Final Answer

$$
\boxed{8}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of f(3) given that f(-3) = 2 for the function f(x) = ax⁴ - bx² + x + 5. Hmm, let me think about how to approach this.

First, let me write down the function again to make sure I have it right: f(x) = a x⁴ - b x² + x + 5. So it's a quartic function with coefficients a and b. The problem gives me f(-3) = 2, and I need to find f(3).

Since the function has both even and odd powers of x, maybe there's some symmetry I can exploit here. Let me recall that for functions involving even and odd powers, sometimes substituting -x can lead to simplifications. Let me try that.

Let me compute f(-x). If I replace x with -x in the function, I get:

f(-x) = a(-x)⁴ - b(-x)² + (-x) + 5.

Calculating each term:

(-x)⁴ is x⁴ because any even power of a negative number is positive. Similarly, (-x)² is x². So substituting back:

f(-x) = a x⁴ - b x² - x + 5.

Now, let me compare this with the original function f(x) = a x⁴ - b x² + x + 5.

So, if I subtract f(-x) from f(x), let's see what happens:

f(x) - f(-x) = [a x⁴ - b x² + x + 5] - [a x⁴ - b x² - x + 5].

Let me compute term by term:

a x⁴ - a x⁴ = 0.

- b x² - (-b x²) = -b x² + b x² = 0.

Then, x - (-x) = x + x = 2x.

And 5 - 5 = 0.

So overall, f(x) - f(-x) = 2x.

That's interesting. So this tells me that f(x) - f(-x) = 2x.

Alternatively, maybe I can express f(x) + f(-x) as well? Let me check:

f(x) + f(-x) = [a x⁴ - b x² + x + 5] + [a x⁴ - b x² - x + 5] = 2a x⁴ - 2b x² + 10.

So that's another relation.

But how does this help me? Well, the problem gives me f(-3) = 2, and I need to find f(3). Let me denote f(3) as some value, say, y. Then, maybe I can set up equations using the relations I found.

Let me write down f(-3) and f(3) using the expressions above.

First, let's compute f(-3):

f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5 = a*(81) - b*(9) - 3 + 5 = 81a - 9b + 2.

And according to the problem, this equals 2. So:

81a - 9b + 2 = 2.

Subtracting 2 from both sides:

81a - 9b = 0.

Simplify this equation. Let me factor out 9:

9*(9a - b) = 0 => 9a - b = 0 => b = 9a.

Okay, so that's one equation relating a and b.

Now, let me compute f(3):

f(3) = a*(3)^4 - b*(3)^2 + 3 + 5 = 81a - 9b + 8.

Wait, so f(3) = 81a - 9b + 8.

But from earlier, we found that 81a - 9b = 0. Therefore, substituting that into f(3):

f(3) = 0 + 8 = 8.

Wait, so is it that straightforward? Let me check again.

Given that f(-3) = 81a - 9b - 3 + 5 = 81a - 9b + 2 = 2. Therefore, 81a - 9b = 0. Then f(3) is 81a - 9b + 3 + 5 = 81a - 9b + 8. Since 81a - 9b is zero, f(3) is 0 + 8 = 8.

So the answer is 8? That seems too straightforward. Let me verify with another approach to make sure I didn't make a mistake.

Alternatively, using the relation f(x) - f(-x) = 2x. Let me plug in x = 3. Then:

f(3) - f(-3) = 2*3 = 6.

We know that f(-3) = 2, so:

f(3) - 2 = 6 => f(3) = 6 + 2 = 8.

Yes! That's the same result. So this confirms that f(3) is 8.

Therefore, regardless of the values of a and b, as long as they satisfy the condition given by f(-3) = 2, the value of f(3) is 8.

Let me check if there's another way to think about this. For example, considering the function's even and odd parts.

The function f(x) can be split into even and odd components. The even part would be the terms with even powers of x, and the odd part would be the terms with odd powers.

Looking at f(x) = ax⁴ - bx² + x + 5.

The even part is ax⁴ - bx² + 5, and the odd part is x.

So, f(x) = [ax⁴ - bx² + 5] + x.

Now, if we consider the even function E(x) = ax⁴ - bx² + 5 and the odd function O(x) = x.

Then, f(-x) = E(-x) + O(-x) = E(x) - O(x).

Therefore, f(-x) = E(x) - O(x).

So, f(x) + f(-x) = 2E(x), and f(x) - f(-x) = 2O(x).

Which is consistent with what I found earlier.

So, since O(x) = x, then f(x) - f(-x) = 2x.

Therefore, if we take x = 3, then f(3) - f(-3) = 6. Since f(-3) = 2, then f(3) = 6 + 2 = 8.

Same result again.

Therefore, regardless of the values of a and b, as long as they satisfy the condition given by f(-3) = 2, the value of f(3) is 8.

Wait, but let me check if there's any possibility that the equations could have more constraints. For example, if I had more information, but here we only have one equation (from f(-3) = 2) and two variables (a and b). However, when calculating f(3), the terms involving a and b cancel out, so we don't need to know their exact values.

Let me verify with actual numbers. Suppose I choose a value for a, then compute b, and check f(3).

For example, let me pick a = 1. Then from b = 9a, we have b = 9.

So then f(x) = x⁴ - 9x² + x + 5.

Compute f(-3):

(-3)^4 - 9*(-3)^2 + (-3) + 5 = 81 - 9*9 - 3 + 5 = 81 - 81 - 3 + 5 = 0 - 3 + 5 = 2.

Which matches the given condition. Now compute f(3):

3^4 - 9*(3)^2 + 3 + 5 = 81 - 81 + 3 + 5 = 0 + 8 = 8.

Perfect, that works.

Let me try another value for a. Let me choose a = 0. Then b = 9*0 = 0.

So f(x) = 0x⁴ - 0x² + x + 5 = x + 5.

Then f(-3) = -3 + 5 = 2, which is correct. Then f(3) = 3 + 5 = 8.

Same result.

Another example: let me take a = 2, so b = 18.

Then f(x) = 2x⁴ - 18x² + x + 5.

Compute f(-3):

2*(81) - 18*(9) + (-3) + 5 = 162 - 162 - 3 + 5 = 0 - 3 + 5 = 2.

Good. Then f(3) = 2*(81) - 18*(9) + 3 + 5 = 162 - 162 + 3 + 5 = 0 + 8 = 8.

Same answer again.

Therefore, regardless of the values of a and b, as long as they satisfy the condition from f(-3) = 2, f(3) is always 8.

Therefore, the answer is 8.

But just to make sure I didn't overlook anything, let me check if there's another way someone might approach this problem and perhaps make a mistake.

For example, someone might think that since f(-3) = 2, then f(3) is something else, maybe related to the coefficients. But since the function has both even and odd terms, the even terms will be the same for x and -x, while the odd terms will have opposite signs.

So, if I write f(3) = even_part(3) + odd_part(3), and f(-3) = even_part(-3) + odd_part(-3) = even_part(3) - odd_part(3).

Therefore, if I add f(3) and f(-3), I get 2*even_part(3). If I subtract f(-3) from f(3), I get 2*odd_part(3).

In this case, the odd_part(3) is just the coefficient of x times 3, which is 1*3 = 3. Therefore, f(3) - f(-3) = 2*3 = 6. Therefore, f(3) = f(-3) + 6 = 2 + 6 = 8.

This is another way to see it, using the even and odd parts.

So, all methods lead to the same conclusion. Therefore, I can be confident that the answer is 8.

I think that's thorough enough. I considered multiple approaches, checked with examples, and verified the relationships between f(x) and f(-x). All roads lead to Rome here.

**Final Answer**
The value of $ f(3) $ is \boxed{8}.

To determine the value of $ f(3) $, we are given the function:

$$
f(x) = ax^4 - bx^2 + x + 5
$$

and the condition:

$$
f(-3) = 2
$$

---

### Step 1: Use the given condition to find a relationship between $ a $ and $ b $

We substitute $ x = -3 $ into the function:

$$
f(-3) = a(-3)^4 - b(-3)^2 + (-3) + 5 = 81a - 9b - 3 + 5 = 81a - 9b + 2
$$

We are told that $ f(-3) = 2 $, so:

$$
81a - 9b + 2 = 2
$$

Subtracting 2 from both sides:

$$
81a - 9b = 0
$$

Divide both sides by 9:

$$
9a - b = 0 \quad \Rightarrow \quad b = 9a
$$

---

### Step 2: Use the relationship to find $ f(3) $

Now compute $ f(3) $:

$$
f(3) = a(3)^4 - b(3)^2 + 3 + 5 = 81a - 9b + 8
$$

Substitute $ b = 9a $:

$$
f(3) = 81a - 9(9a) + 8 = 81a - 81a + 8 = 8
$$

---

### Step 3: Confirm using symmetry

We can also observe that the function has both even and odd components:

- Even part: $ ax^4 - bx^2 + 5 $
- Odd part: $ x $

So, the function can be written as:

$$
f(x) = \text{even}(x) + \text{odd}(x)
$$

Then:

$$
f(-x) = \text{even}(x) - \text{odd}(x)
$$

So:

$$
f(x) - f(-x) = 2 \cdot \text{odd}(x) = 2x
$$

This gives us:

$$
f(3) - f(-3) = 2 \cdot 3 = 6
$$

Since $ f(-3) = 2 $, we find:

$$
f(3) = 2 + 6 = 8
$$

---

### Final Answer

$$
\boxed{8}
$$

Solution: Evaluating $f(x)$ for $x=3$ and $x=-3$, we have \[\left\{ \begin{aligned} f(3)& = a \cdot 3^4 - b \cdot 3^2 + 3 + 5, \\ f(-3) &= a \cdot (-3)^4 - b \cdot (-3)^2 + (-3) + 5. \end{aligned} \right.\]If we subtract the second equation from the first equation, all the terms but one cancel out, and we get \[f(3) - f(-3) = 3 - (-3) = 6.\]Thus, if $f(-3) = 2,$ then $f(3) = f(-3) + 6 = 2 + 6 = \boxed{8}.$


Explanation:
Okay, so I need to find the equation of a hyperbola given two foci, F₁ = (10, 2) and F₂ = (-16, 2), and the condition that the absolute difference of the distances from any point P on the hyperbola to the two foci is 24. Then, once I have the equation in the standard form, I need to find h + k + a + b. Alright, let me recall what I know about hyperbolas.

First, a hyperbola is defined as the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant. That constant is usually denoted as 2a, where a is the distance from the center to a vertex. So in this case, the given condition |PF₁ - PF₂| = 24 implies that 2a = 24, so a = 12. Got that.

Next, I need to find the center of the hyperbola. The center is the midpoint between the two foci. Let me calculate that. The midpoint formula is ((x₁ + x₂)/2, (y₁ + y₂)/2). Plugging in the coordinates of F₁ and F₂:

Midpoint h, k = ((10 + (-16))/2, (2 + 2)/2) = ((-6)/2, 4/2) = (-3, 2). So the center is at (-3, 2). Therefore, h = -3 and k = 2. So far, so good.

Now, I need to figure out the values of a, b, and the orientation of the hyperbola. Wait, we already found a = 12. Now, the distance between the two foci is 2c, where c is the distance from the center to each focus. Let me compute that. The distance between F₁ and F₂ can be found using the distance formula. Since both foci have the same y-coordinate, 2, the distance between them is just the difference in the x-coordinates. So distance = |10 - (-16)| = |26| = 26. Therefore, 2c = 26, so c = 13.

Now, for hyperbolas, the relationship between a, b, and c is c² = a² + b². So once I know a and c, I can solve for b. Let me write that down:

c² = a² + b²
=> b² = c² - a²
=> b² = 13² - 12²
=> b² = 169 - 144
=> b² = 25
=> b = 5

So b is 5. Therefore, the standard form of the hyperbola equation is:

[(x - h)² / a²] - [(y - k)² / b²] = 1

But wait, I need to make sure that the hyperbola is oriented correctly. Since the foci are located along the horizontal line y = 2, which is horizontal, the transverse axis is horizontal. Therefore, the hyperbola opens left and right, which means the standard form is indeed [(x - h)² / a²] - [(y - k)² / b²] = 1. So that's correct.

Let me double-check the center. The midpoint between (-16, 2) and (10, 2) is indeed (-3, 2). So h = -3, k = 2. Then a = 12, b = 5. Therefore, the equation is:

[(x - (-3))² / 12²] - [(y - 2)² / 5²] = 1
Which simplifies to:
[(x + 3)² / 144] - [(y - 2)² / 25] = 1

So the question asks for h + k + a + b. Let me compute that:

h = -3
k = 2
a = 12
b = 5
So adding them up: (-3) + 2 + 12 + 5 = (-3 + 2) + (12 + 5) = (-1) + 17 = 16.

Wait, that seems straightforward, but let me check again to make sure I didn't make any mistakes. Let me verify each step.

First, the midpoint: (10 + (-16))/2 = (-6)/2 = -3, and (2 + 2)/2 = 2. Correct. So center at (-3, 2). Then, distance between foci: 10 - (-16) is 26, so 2c = 26, c = 13. Then, since |PF₁ - PF₂| = 2a = 24, so a = 12. Then c² = a² + b² => 169 = 144 + b² => b² = 25 => b = 5. All correct.

Then, the standard form equation is [(x + 3)^2]/144 - [(y - 2)^2]/25 = 1. Therefore, h = -3, k = 2, a = 12, b = 5. Adding them: -3 + 2 + 12 + 5. Let me compute again:

-3 + 2 is -1. Then 12 + 5 is 17. Then -1 + 17 is 16. So the answer is 16. That seems right.

But let me check if the hyperbola is indeed horizontal. Since the foci are on the same horizontal line, the transverse axis is horizontal, so yes, the hyperbola opens left and right, so the standard form is correct. If the foci were on a vertical line, the equation would have y terms first. But here, since they are on a horizontal line, x terms first. So that's correct.

Another thing to check: the value of a. Since the difference of distances is 2a, and that's given as 24, so a is 12. Correct. And c is 13, which is greater than a, which is necessary for hyperbola. So that's okay.

Therefore, I think my answer is correct. The sum h + k + a + b is 16.

**Final Answer**
\boxed{16}

To solve the problem, we are given two foci of a hyperbola:
$$
F_1 = (10, 2), \quad F_2 = (-16, 2)
$$
and the condition that for any point $ P $ on the hyperbola, the absolute difference of distances to the foci is constant:
$$
|PF_1 - PF_2| = 24
$$
This is the defining property of a hyperbola, where the constant difference is $ 2a $, so we can immediately deduce:
$$
2a = 24 \Rightarrow a = 12
$$

---

### Step 1: Find the Center of the Hyperbola

The center of the hyperbola is the midpoint of the segment joining the two foci. Using the midpoint formula:
$$
\text{Center} = \left( \frac{10 + (-16)}{2}, \frac{2 + 2}{2} \right) = \left( \frac{-6}{2}, \frac{4}{2} \right) = (-3, 2)
$$
So, the center is at $ (h, k) = (-3, 2) $.

---

### Step 2: Determine the Value of $ c $

The distance between the foci is:
$$
\text{Distance} = |10 - (-16)| = 26
$$
This is equal to $ 2c $, so:
$$
2c = 26 \Rightarrow c = 13
$$

---

### Step 3: Use the Hyperbola Identity to Find $ b $

For a hyperbola, the relationship between $ a $, $ b $, and $ c $ is:
$$
c^2 = a^2 + b^2
$$
Substituting the known values:
$$
13^2 = 12^2 + b^2 \Rightarrow 169 = 144 + b^2 \Rightarrow b^2 = 25 \Rightarrow b = 5
$$

---

### Step 4: Write the Equation of the Hyperbola

Since the foci lie on a horizontal line (same y-coordinate), the transverse axis is horizontal. Therefore, the standard form of the hyperbola is:
$$
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
$$
Substituting the values:
$$
\frac{(x + 3)^2}{144} - \frac{(y - 2)^2}{25} = 1
$$

---

### Step 5: Compute $ h + k + a + b $

From earlier:
- $ h = -3 $
- $ k = 2 $
- $ a = 12 $
- $ b = 5 $

So:
$$
h + k + a + b = -3 + 2 + 12 + 5 = 16
$$

---

### Final Answer

$$
\boxed{16}
$$

[Assistant]
Explanation:
Okay, so I need to find the equation of a hyperbola given two foci, F₁ = (10, 2) and F₂ = (-16, 2), and the condition that the absolute difference of the distances from any point P on the hyperbola to the two foci is 24. Then, once I have the equation in the standard form, I need to find h + k + a + b. Alright, let me recall what I know about hyperbolas.

First, a hyperbola is defined as the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant. That constant is usually denoted as 2a, where a is the distance from the center to a vertex. So in this case, the given condition |PF₁ - PF₂| = 24 implies that 2a = 24, so a = 12. Got that.

Next, I need to find the center of the hyperbola. The center is the midpoint between the two foci. Let me calculate that. The midpoint formula is ((x₁ + x₂)/2, (y₁ + y₂)/2). Plugging in the coordinates of F₁ and F₂:

Midpoint h, k = ((10 + (-16))/2, (2 + 2)/2) = ((-6)/2, 4/2) = (-3, 2). So the center is at (-3, 2). Therefore, h = -3 and k = 2. So far, so good.

Now, I need to figure out the values of a, b, and the orientation of the hyperbola. Wait, we already found a = 12. Now, the distance between the two foci is 2c, where c is the distance from the center to each focus. Let me compute that. The distance between F₁ and F₂ can be found using the distance formula. Since both foci have the same y-coordinate, 2, the distance between them is just the difference in the x-coordinates. So distance = |10 - (-16)| = |26| = 26. Therefore, 2c = 26, so c = 13.

Now, for hyperbolas, the relationship between a, b, and c is c² = a² + b². So once I know a and c, I can solve for b. Let me write that down:

c² = a² + b²
=> b² = c² - a²
=> b² = 13² - 12²
=> b² = 169 - 144
=> b² = 25
=> b = 5

So b is 5. Therefore, the standard form of the hyperbola equation is:

[(x - h)² / a²] - [(y - k)² / b²] = 1

But wait, I need to make sure that the hyperbola is oriented correctly. Since the foci are located along the horizontal line y = 2, which is horizontal, the transverse axis is horizontal. Therefore, the hyperbola opens left and right, which means the standard form is indeed [(x - h)² / a²] - [(y - k)² / b²] = 1. So that's correct.

Let me double-check the center. The midpoint between (-16, 2) and (10, 2) is indeed (-3, 2). So h = -3, k = 2. Then a = 12, b = 5. Therefore, the equation is:

[(x - (-3))² / 12²] - [(y - 2)² / 5²] = 1
Which simplifies to:
[(x + 3)² / 144] - [(y - 2)² / 25] = 1

So the question asks for h + k + a + b. Let me compute that:

h = -3
k = 2
a = 12
b = 5
So adding them up: (-3) + 2 + 12 + 5 = (-3 + 2) + (12 + 5) = (-1) + 17 = 16.

Wait, that seems straightforward, but let me check again to make sure I didn't make any mistakes. Let me verify each step.

First, the midpoint: (10 + (-16))/2 = (-6)/2 = -3, and (2 + 2)/2 = 2. Correct. So center at (-3, 2). Then, distance between foci: 10 - (-16) is 26, so 2c = 26, c = 13. Then, since |PF₁ - PF₂| = 2a = 24, so a = 12. Then c² = a² + b² => 169 = 144 + b² => b² = 25 => b = 5. All correct.

Then, the standard form equation is [(x + 3)^2]/144 - [(y - 2)^2]/25 = 1. Therefore, h = -3, k = 2, a = 12, b = 5. Adding them: -3 + 2 + 12 + 5. Let me compute again:

-3 + 2 is -1. Then 12 + 5 is 17. Then -1 + 17 is 16. So the answer is 16. That seems right.

But let me check if the hyperbola is indeed horizontal. Since the foci are on the same horizontal line, the transverse axis is horizontal, so yes, the hyperbola opens left and right, so the standard form is correct. If the foci were on a vertical line, the equation would have y terms first. But here, since they are on a horizontal line, x terms first. So that's correct.

Another thing to check: the value of a. Since the difference of distances is 2a, and that's given as 24, so a is 12. Correct. And c is 13, which is greater than a, which is necessary for hyperbola. So that's okay.

Therefore, I think my answer is correct. The sum h + k + a + b is 16.

**Final Answer**
\boxed{16}

To solve the problem, we are given two foci of a hyperbola:
$$
F_1 = (10, 2), \quad F_2 = (-16, 2)
$$
and the condition that for any point $ P $ on the hyperbola, the absolute difference of distances to the foci is constant:
$$
|PF_1 - PF_2| = 24
$$
This is the defining property of a hyperbola, where the constant difference is $ 2a $, so we can immediately deduce:
$$
2a = 24 \Rightarrow a = 12
$$

---

### Step 1: Find the Center of the Hyperbola

The center of the hyperbola is the midpoint of the segment joining the two foci. Using the midpoint formula:
$$
\text{Center} = \left( \frac{10 + (-16)}{2}, \frac{2 + 2}{2} \right) = \left( \frac{-6}{2}, \frac{4}{2} \right) = (-3, 2)
$$
So, the center is at $ (h, k) = (-3, 2) $.

---

### Step 2: Determine the Value of $ c $

The distance between the foci is:
$$
\text{Distance} = |10 - (-16)| = 26
$$
This is equal to $ 2c $, so:
$$
2c = 26 \Rightarrow c = 13
$$

---

### Step 3: Use the Hyperbola Identity to Find $ b $

For a hyperbola, the relationship between $ a $, $ b $, and $ c $ is:
$$
c^2 = a^2 + b^2
$$
Substituting the known values:
$$
13^2 = 12^2 + b^2 \Rightarrow 169 = 144 + b^2 \Rightarrow b^2 = 25 \Rightarrow b = 5
$$

---

### Step 4: Write the Equation of the Hyperbola

Since the foci lie on a horizontal line (same y-coordinate), the transverse axis is horizontal. Therefore, the standard form of the hyperbola is:
$$
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
$$
Substituting the values:
$$
\frac{(x + 3)^2}{144} - \frac{(y - 2)^2}{25} = 1
$$

---

### Step 5: Compute $ h + k + a + b $

From earlier:
- $ h = -3 $
- $ k = 2 $
- $ a = 12 $
- $ b = 5 $

So:
$$
h + k + a + b = -3 + 2 + 12 + 5 = 16
$$

---

### Final Answer

$$
\boxed{16}
$$

Solution:
The center of the hyperbola is the midpoint of $\overline{F_1 F_2},$ which is $(-3,2).$  Thus, $h = -3$ and $k = 2.$

Also, $2a = 24,$ so $a = 12.$  The distance between the foci is $2c = 26,$ so $c = 13.$  Then $b^2 = c^2 - a^2 = 169 - 144 = 25,$ so $b = 5.$

Hence, $h + k + a + b = (-3) + 2 + 12 + 5 = \boxed{16}.$