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

Set up the conditions.

Let the center of the circle be $$(h, k)$$ with radius $$r$$.

Since the circle touches the $$y$$-axis, the distance from the center to the $$y$$-axis equals the radius:

$$r = |h|$$

Since the circle also touches the line $$x + y = 0$$, the distance from the center to this line equals the radius:

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

Solve for the relationship.

Squaring both sides:

$$\frac{(h + k)^2}{2} = h^2$$

$$(h + k)^2 = 2h^2$$

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

$$k^2 + 2hk - h^2 = 0$$

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

Write the locus by replacing $$h$$ with $$x$$ and $$k$$ with $$y$$:

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

Therefore, the locus of the center is $$x^2 - y^2 = 2xy$$, which is Option D.