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The problem you uploaded involves finding the coordinates of point \( P \) and the area of triangle \( ABP \), given the points \( A(-4, 0) \) and \( B(4, 0) \) with the condition that:

\[
AP^2 + BP^2 = 64
\]

Let's break down the solution step-by-step:

### Step 1: Express \( AP^2 \) and \( BP^2 \) 
Let the coordinates of \( P \) be \( (x, y) \).

- The distance from \( A(-4, 0) \) to \( P(x, y) \) is:
  \[
  AP = \sqrt{(x + 4)^2 + y^2}
  \]
  Hence,
  \[
  AP^2 = (x + 4)^2 + y^2
  \]

- The distance from \( B(4, 0) \) to \( P(x, y) \) is:
  \[
  BP = \sqrt{(x - 4)^2 + y^2}
  \]
  Hence,
  \[
  BP^2 = (x - 4)^2 + y^2
  \]

### Step 2: Use the given condition
We are told that:
\[
AP^2 + BP^2 = 64
\]
Substituting the expressions for \( AP^2 \) and \( BP^2 \):
\[
(x + 4)^2 + y^2 + (x - 4)^2 + y^2 = 64
\]

### Step 3: Simplify the equation
Expanding both squared terms:
\[
(x + 4)^2 = x^2 + 8x + 16
\]
\[
(x - 4)^2 = x^2 - 8x + 16
\]
Substituting these into the equation:
\[
(x^2 + 8x + 16) + y^2 + (x^2 - 8x + 16) + y^2 = 64
\]
Combine like terms:
\[
2x^2 + 2y^2 + 32 = 64
\]
Simplifying further:
\[
x^2 + y^2 + 16 = 32
\]
\[
x^2 + y^2 = 16
\]
This is the equation of a circle with center at the origin \( (0, 0) \) and radius 4. Therefore, point \( P \) lies on this circle.

### Step 4: Area of the triangle
The area \( A \) of triangle \( ABP \) can be calculated using the formula for the area of a triangle given its vertices:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substitute \( A(-4, 0) \), \( B(4, 0) \), and \( P(x, y) \) into the formula to get the area once we know the exact coordinates of \( P \).

Would you like me to proceed with calculating the possible coordinates of \( P \) and the area of the triangle?

---

**5 Relative Questions:**
1. How do you find the equation of a circle from given coordinates?
2. Can you explain how to derive the distance formula for two points in a plane?
3. Why does the equation simplify into the formula of a circle in this case?
4. How does knowing the radius of a circle help in determining point coordinates?
5. What is the geometric interpretation of \( AP^2 + BP^2 = 64 \)?

**Tip:** Remember that simplifying equations involving sums of squares often leads to conic sections, such as circles or ellipses.

Diketahui sebuah segitiga ABP dengan titik sudut A(-4,0) dan B(4,0), Jika AP^2 + BP^2 = 64, maka temukan titik koordinat P dan luas segitiganya.

Solve the problem of finding the coordinates of point P and the area of triangle ABP with given points A(-4,0) and B(4,0) under the condition AP^2 + BP^2 = 64. This problem covers key mathematical concepts including the distance formula and the equation of a circle.

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