A circle touches both the y -axis and the line x + y = 0 . Then the...

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

$⇒$$\left|h\right| = \dfrac{\left|h + k\right|}{\sqrt{2}}$

squaring both sides we get

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

$⇒ 2{h}^{2} = {h}^{2} + {k}^{2} + 2hk$

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

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

$⇒ {x}^{2} - {y}^{2} = 2xy$