Question

Find the locus of a point which moves such that its distance from the origin is three times its distance from $x$-axis.

Answer 1

Let $p(h,k)$ be any point on the locus and let $O(0,0)$ be the origin.

By the given condition $OP=3k$ [$\because$ $k$ is the difference of point from $x$ axis]

$\Rightarrow$ $OP$${}^2$ = $9$$K^2$

$\Rightarrow \sqrt{(0-h{)}^2+(0-k{)}^2}$ = $9k$${}^2$

$\Rightarrow$ $h$${}^2$+$k$${}^2$ = $9k$${}^2$

$\Rightarrow$ $h$${}^2$ = $9k$${}^2$- $k$${}^2$

$\Rightarrow$ $h$${}^2$=$8k$${}^2$

Hence, locusof$(h,k)$is$x$${}^2$ = $8y$${}^2$ = 0