Question

Find the equation of the curve formed by the set of all those points the sum of whose distances from the points A(4, 0, 0) and B(-4, 0, 0) is 10 units.

Answer 1

Let P(x, y, z) be an arbitrary point on the given curve.

Then,

PA+PB=10

$\Rightarrow \sqrt{(x-4{)}^2+y^2+z^2}+\sqrt{(x+4{)}^2+y^2+z^2}=10$

$\Rightarrow \sqrt{(x+4{)}^2+y^2+z^2}=10-\sqrt{(x-4{)}^2+y^2+z^2}$ ...(i)

$\Rightarrow (x+4{)}^2+y^2+z^2=100+(x-4{)}^2+y^2+z^2-20\sqrt{(x-4{)}^2+y^2+z^2}$ [On squaring both sides of (i)]

$16x=100-20\sqrt{(x-4{)}^2+y^2+z^2}$

$\Rightarrow 5\sqrt{(x-4{)}^2+y^2+z^2}=(25-4x)$

$\Rightarrow 25\left[\right(x-4{)}^2+y^2+z^2]=625+16x^2-200x$

$\Rightarrow 9x^2+25y^2+25z^2-225=0$

Hence, the required equation of the curve is

$9x^2+25y^2+25z^2-225=0$.