Question

A point moves such that its distance from the point $(4,0)$ is half that of its distance from the line $x=16$, find its locus.

Answer 1

Let $P(h,k)$ be the variable point.

By hypothesis,

$\sqrt{(h-4{)}^2+(k-0{)}^2}=\frac{1}{2}\left|\frac{h-16}{\sqrt{1^2+0^2}}\right|$

$4\left[\right(h-4{)}^2+k^2]=(h-16{)}^2$

$3h^2+4k^2=192$

Hence, the locus of $(h,k)$ is $3x^2+4y^2=192$.