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(D) Let the center of the circle be $(h, k)$ and its radius be $r$.Since the circle touches the $y$-axis $(x=0)$,the radius $r = |h|$.Since the circle

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

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(D) Let the center of the circle be $(h, k)$ and its radius be $r$.Since the circle touches the $y$-axis $(x=0)$,the radius $r = |h|$.Since the circle also touches the line $x+y=0$,the perpendicular distance from the center $(h, k)$ to the line $x+y=0$ must be equal to the radius $r$.Thus,$r = \frac{|h+k|}{\sqrt{1^{2}+1^{2}}} = \frac{|h+k|}{\sqrt{2}}$.Equating the two expressions for $r$,we get $|h| = \frac{|h+k|}{\sqrt{2}}$.Squaring both sides,we have $h^{2} = \frac{(h+k)^{2}}{2}$.$2h^{2} = h^{2} + k^{2} + 2hk$.$h^{2} - k^{2} = 2hk$.Replacing $(h, k)$ with $(x, y)$,the locus of the center is $x^{2}-y^{2}=2xy$.

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

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