Question

Find the locus of a point whose distance from the point $(1,2)$ is equal to its distance from the axis of $y$.

Answer 1

Let the point be $P(h,k)$ and $A$ be $(1,2)$

Equation of $y$ axis is $x=0$

Distance of $P$ from $y$ axis be $PB$

Given $PA=PB$

$PA=\sqrt{{(h-1)}^2+{(k-2)}^2}\sqrt{{(h-1)}^2+{(k-2)}^2}=h^2$

Squaring both sides

${(h-1)}^2+{(k-2)}^2=h^{2}1-2h+k^2+4-4k=1k^2-2h-4k+5=0$

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

$2x+y^2-4y+5=0$