A circle touches both the $y$-axis and the line $x+y=0$. Then the locus of its center

Plotting the diagram of given data we have,

Now to find locus, let centre be $\left(h,k\right)$. According to question

$\left|OM\right|=\left|ON\right|$ distance 

$\Rightarrow$$\left|h\right|=\frac{\left|h+k\right|}{\sqrt{2}}$

squaring both sides we get

$\Rightarrow 2h^2={\left(h+k\right)}^2$  

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

Replacing $h$ with $x$ & $k$ with $y$ we get

$2x^2=x^2+y^2+2xy$

$\Rightarrow x^2-y^2=2xy$