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 $P(x,y,z)$ be the point such that

$PA+PB=10$

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

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

Now squaring both sides we get,

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

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

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

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

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

Again squaring both sides we get,

or, $16x^2-200x+625=25\left\{\right(x^2-8x+16)+y^2+z^2\}$

or, $9x^2+25y^2+25z^2=225$.

This is the required equation.