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

• A

  $y=\sqrt{2} x$
• B

  $x=\sqrt{2} y$
• C

  $y^2-x^2=2 x y$
• Correct answer
  D

  $x^2-y^2=2 x y$

Solution

Let $(h, k)$ is centre of circle
$\left|\frac{h-k}{\sqrt{2}}\right|=|h|, k^2-h^2+2 h k=0$

Hence, the correct answer is option: 'D'.