Question

Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (-4, 0, 0) is equal to 10.

Answer 1

Let the coordinates of P be (x, y, z).
The coordinates of points A and B are (4, 0, 0) and (- 4, 0, 0) respectively.
It is given that PA + PB = 10.
⇒\(\sqrt{(x-4)^2+y^2+z^2}+\sqrt{(x+4)^2+y^2+z^2}\) = 10
⇒\(\sqrt{(x-4)^2+y^2+z^2}\) = 10 − \(\sqrt{(x+4)^2+y^2+z^2}\)
On squaring both sides, we obtain
⇒ \({(x-4)^2+y^2+z^2}\) = \(100-20\sqrt{(x+4)^2+y^2+z^2+(x+4)^2+y^2+z^2}\)
⇒\(x^2-8x+16+y^2+z^2\) = \(100-20\sqrt{x^2+8x+16+y^2+z^2+x^2+8x+16+y^2+z^2}\)
⇒ 5\(\sqrt{x^2+8x+16+y^2+z^2}\) = (25+4x)
On squaring both sides again, we obtain
25 (\(\sqrt{x^2+8x+16+y^2+z^2}\)) = 625 +16x² + 200x
⇒ 9x² + 25y² + 25z² - 225 = 0
Thus, the required equation is 9x²+ 25y²+25z - 225 = 0.