Let A (4, 0, 0) & B (– 4, 0, 0)

Let the coordinates of point P be (x, y, z)

Calculating PA

P ≡ (x, y, z) and A ≡ (4, 0, 0)

Distance PA

Here,

x₁ = x, y₁ = y, z₁ = z

x₂ = 4, y₂ = 0, z₂ = 0

Distance PA

Calculating PB

P ≡ (x, y, z) and B ≡ (– 4, 0, 0)

Distance PB

Here,

x₁ = x, y₁ = y, z₁ = z

x₂ = – 4, y₂ = 0, z₂ = 0

Distance PB

Given that –

PA + PB = 10

⇒ PA = 10 – PB

Squaring both sides, we get –

PA² = (10 – PB)²

⇒ PA² = 100 + PB² – 20 PB

⇒ (4 – x)² + (0 – y)² + (0 – z)²

= 100 + (– 4 – x)² + (0 – y)² + (0 – z)² – 20 PB

⇒ (16 + x² – 8x) + (y²) + (z²)

= 100 + (16 + x² + 8x) + (y²) + (z²) – 20 PB

⇒ 20 PB = 16x + 100

⇒ 5 PB = (4x + 25)

Squaring both sides again, we get –

⇒ 25 PB² = 16x² + 200x + 625

⇒ 25 [(– 4 – x)² + (0 – y)² + (0 – z)²] = 16x² + 200x + 625

⇒ 25 [x² + y² + z² + 8x + 16] = 16x² + 200x + 625

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

⇒ 9x² + 25y² + 25z² – 225 = 0

This is the required equation.