Question

Find the equation to the locus of a point which moves so that the square of its distance from the point $(0,2)$ is equal to $4$.

Answer 1

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

Given point be $A(0,2)$

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

Given ${PA}^2=4$

${(h-0)}^2+{(k-2)}^2=4h^2+k^2+4-4k=4h^2+k^2-4k=0$

Replacing $h$ by $x$ and $k$ by $y$

$x^2+y^2=4y$

is the required locus.