Find the equation of the set of points P, the sum of whose distances from

A(4,0,0) & B(−4,0,0) is equal to 10

The problem describes a set of points \( P(x, y, z) \) such that the sum of the distances from \( P \) to two fixed points \( A(4, 0, 0) \) and \( B(-4, 0, 0) \) is constant. This is a classic definition of an **ellipse** in three-dimensional space.

### Solution:

Let the distance from \( P(x, y, z) \) to \( A(4, 0, 0) \) be \( d_A \) and to \( B(-4, 0, 0) \) be \( d_B \). We are given:

\[
d_A + d_B = 10
\]

The distance from \( P(x, y, z) \) to \( A(4, 0, 0) \) is:

\[
d_A = \sqrt{(x - 4)^2 + y^2 + z^2}
\]

The distance from \( P(x, y, z) \) to \( B(-4, 0, 0) \) is:

\[
d_B = \sqrt{(x + 4)^2 + y^2 + z^2}
\]

So, the equation we need to solve is:

\[
\sqrt{(x - 4)^2 + y^2 + z^2} + \sqrt{(x + 4)^2 + y^2 + z^2} = 10
\]

### Squaring both sides:

To eliminate the square roots, square both sides of the equation:

\[
\left(\sqrt{(x - 4)^2 + y^2 + z^2} + \sqrt{(x + 4)^2 + y^2 + z^2}\right)^2 = 10^2
\]

Expanding the square:

\[
(x - 4)^2 + y^2 + z^2 + (x + 4)^2 + y^2 + z^2 + 2\sqrt{\left((x - 4)^2 + y^2 + z^2\right)\left((x + 4)^2 + y^2 + z^2\right)} = 100
\]

Simplify the equation:

\[
(x - 4)^2 + (x + 4)^2 + 2y^2 + 2z^2 + 2\sqrt{\left((x - 4)^2 + y^2 + z^2\right)\left((x + 4)^2 + y^2 + z^2\right)} = 100
\]

Expanding \( (x - 4)^2 \) and \( (x + 4)^2 \):

\[
(x^2 - 8x + 16) + (x^2 + 8x + 16) + 2y^2 + 2z^2 + 2\sqrt{\left((x - 4)^2 + y^2 + z^2\right)\left((x + 4)^2 + y^2 + z^2\right)} = 100
\]

Simplify:

\[
2x^2 + 32 + 2y^2 + 2z^2 + 2\sqrt{\left((x - 4)^2 + y^2 + z^2\right)\left((x + 4)^2 + y^2 + z^2\right)} = 100
\]

Subtracting 64 from both sides:

\[
2x^2 + 2y^2 + 2z^2 + 2\sqrt{\left((x - 4)^2 + y^2 + z^2\right)\left((x + 4)^2 + y^2 + z^2\right)} = 36
\]

Divide the entire equation by 2:

\[
x^2 + y^2 + z^2 + \sqrt{\left((x - 4)^2 + y^2 + z^2\right)\left((x + 4)^2 + y^2 + z^2\right)} = 18
\]

### Squaring Again:

Now square both sides again:

\[
\left(x^2 + y^2 + z^2 - 18\right)^2 = \left(\left((x - 4)^2 + y^2 + z^2\right)\left((x + 4)^2 + y^2 + z^2\right)\right)
\]

This equation represents the locus of points \( P(x, y, z) \) for which the sum of the distances to the foci is constant, which defines an ellipse in 3D.

Would you like more details or have any questions?

### Relative Questions:

1. How does the derived equation represent an ellipse in 3D space?
2. What happens to the equation if the sum of distances is less than the given constant?
3. Can you generalize this process for other fixed points in space?
4. What are the geometrical properties of this ellipse?
5. How does the major axis of the ellipse relate to the given constant distance?

### Tip:

When solving equations involving distances, consider simplifying the square root terms by squaring them carefully to avoid unnecessary complexity.

Learn how to find the equation of the set of points P in 3D space that are equidistant from two fixed points A(4, 0, 0) and B(-4, 0, 0). This problem involves the geometry of ellipses in 3D space and the distance formula. Suitable for advanced high school students and beyond.

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