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(C) Given the line $x + y = 0$ $(i)$ and the circle $x^2 + y^2 + 4y = 0$ $(ii)$.Substitute $x = -y$ from $(i)$ into $(ii)$:$(-y)^2 + y^2 + 4y = 0$$2y^

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$y^2 = 2x$

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(C) Given the line $x + y = 0$ $(i)$ and the circle $x^2 + y^2 + 4y = 0$ $(ii)$.Substitute $x = -y$ from $(i)$ into $(ii)$:$(-y)^2 + y^2 + 4y = 0$$2y^2 + 4y = 0$$2y(y + 2) = 0$So,$y = 0$ or $y = -2$.If $y = 0$,then $x = 0$. Point is $(0, 0)$.If $y = -2$,then $x = 2$. Point is $(2, -2)$.Now,check the options for a parabola passing through $(0, 0)$ and $(2, -2)$:For $y^2 = 2x$:At $(0, 0)$: $0^2 = 2(0) \implies 0 = 0$ (Satisfied).At $(2, -2)$: $(-2)^2 = 2(2) \implies 4 = 4$ (Satisfied).Thus,the correct equation is $y^2 = 2x$.

The equation of a parabola which passes through the intersection points of the straight line $x + y = 0$ and the circle $x^2 + y^2 + 4y = 0$ is:

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