Let P(x, y, z) be any point. Then PA = $\sqrt{(x-4)^2+(y-0)^2+(z-0)^2}$ = $\sqrt{x^2+16-8x+y^2+z^2}$ PB = $\sqrt{(x+4)^2+(y-0)^2+(z-0)^2}$ = $\sqrt{x^2+16+8x+y^2+z^2}$ It is given that PA + PB = 10 ∴ $\sqrt{x^2+16-8x+y^2+z^2}$ + $\sqrt{x^2+16+8x+y^2+z^2}$ = 10 ⇒ $\sqrt{x^2+16-8x+y^2+z^2}$ = 10 - $\sqrt{x^2+16+8x+y^2+z^2}$ ... (i) Squaring (i) on both sides, we have $x^2$ + 16 - 8x + $y^2+z^2$ = 100 + $x^2$ + 16 + 8x + $y^2+z^2$ - 20 $\sqrt{x^2+16+8x+y^2+z^2}$ ⇒ 20 $\sqrt{x^2+16+8x+y^2+z^2}$ = 16x + 100 ⇒ 5 = 4x + 25 $\sqrt{x^2+16+8x+y^2+z^2}$ Squaring on both sides again, we have 25 ($x^2 + 16 + 8x + y^2 + z^2$) = $16x^2$ + 625 + 200x ⇒ $25x^2$ + 400 + 200x + $25y^2 + 25z^2 - 16x^2$ – 625 – 200x = 0 ⇒ $9x^2 + 25y^2 + 25z^2$ – 225 = 0 Thus the required equation is $9x^2 + 25y^2 + 25z^2$ – 225 = 0