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Question

Mathematics Question on distance between two points

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

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.
⇒(x−4)2+y2+z2+(x+4)2+y2+z2\sqrt{(x-4)^2+y^2+z^2}+\sqrt{(x+4)^2+y^2+z^2} = 10
⇒(x−4)2+y2+z2\sqrt{(x-4)^2+y^2+z^2} = 10 − (x+4)2+y2+z2\sqrt{(x+4)^2+y^2+z^2}
On squaring both sides, we obtain
⇒ (x−4)2+y2+z2{(x-4)^2+y^2+z^2} = 100−20(x+4)2+y2+z2+(x+4)2+y2+z2100-20\sqrt{(x+4)^2+y^2+z^2+(x+4)^2+y^2+z^2}
⇒x2−8x+16+y2+z2x^2-8x+16+y^2+z^2 = 100−20x2+8x+16+y2+z2+x2+8x+16+y2+z2100-20\sqrt{x^2+8x+16+y^2+z^2+x^2+8x+16+y^2+z^2}
⇒ 5x2+8x+16+y2+z2\sqrt{x^2+8x+16+y^2+z^2} = (25+4x)
On squaring both sides again, we obtain
25 (x2+8x+16+y2+z2\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.