Given: Points are A(4, 0, 0) and B(-4, 0, 0)

To find: the locus of point P, the sum of whose distances from the given points is equal to 10, i.e. PA + PB = 10

Let the required point P(x, y, z)

Formula used:

The distance between any two points (a, b, c) and (m, n, o) is given by,

Therefore,

The distance between P(x, y, z) and A(4, 0, 0) is PA,

Distance between P(x, y, z) and B(-4, 0, 0) is PB,

According to question:

PA + PB = 10

Squaring both sides:

Squaring both sides:

⇒ 16x²+ 625 – 100x = 25x²+ 400 + 200x + 25y² + 25z²

⇒ 16x²+ 625 – 100x – 25x²– 400 – 200x – 25y² – 25z² = 0

⇒ -9x² – 25y² – 25z² – 300x + 225 = 0

⇒ 9x² + 25y² + 25z² + 300x – 225 = 0

Hence locus of point P is 9x² + 25y² + 25z² + 300x – 225 = 0