JEE Main · 2022 · 25 Jun 2022 · Shift 2 · Circle

Let the center of the circle be $(h,k)$. Since the circle touches both the y-axis and the line $x+y=0$, the distances from the center to these two lines must be equal (both equal to the radius).

Distance from $(h,k)$ to y-axis: $|h|$

Distance from $(h,k)$ to line $x+y=0$: $\frac{|h+k|}{\sqrt{2}}$

Equating: $|h|=\frac{|h+k|}{\sqrt{2}}$

Squaring: $h^2=\frac{(h+k{)}^2}{2}\Rightarrow 2h^2=(h+k{)}^2\Rightarrow 2h^2=h^2+2hk+k^2$

$\Rightarrow h^2-k^2=2hk$

The locus is $x^2-y^2=2xy$.