Explanation

(c)
Given that focus is $S(0,\ 0)$
Let $A$ is the vertex of parabola. Take any point $Z$ on the directrix such that $AS = AZ$. Since, the given tangent $x - y + 1 = 0$ is parallel to the directrix
Equation of directrix is $x - y + \lambda = 0$
$\because A$ is the mid point of $SZ$
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$\therefore SZ = 2SA$
$\Rightarrow \frac{|0 - 0 + \lambda|}{\sqrt{1^{2} + 1^{2}}} = 2 \times \frac{|0 - 0 + 1|}{\sqrt{1^{2} + 1^{2}}}$
$\Rightarrow |\lambda| = 2 \Rightarrow \lambda = 2$
$\therefore$ Equation of directrix is $x - y + 2 = 0$
Now, $P$ be any point on the parabola
$\therefore SP = PM \Rightarrow {SP}^{2} = {PM}^{2}$
$\Rightarrow {(x - 0)}^{2} + {(y - 0)}^{2} = \left( \frac{|x - y + 2|}{\sqrt{2}} \right)^{2}$
$\Rightarrow x^{2} + y^{2} + 2xy - 4x + 4y - 4 = 0$