$The equation of the parabola whose focus is the point (0, 0) and the tangent at the vertex is x - y + 1 = 0 is$

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$x^{2} + y^{2} - 2 x y + 4 x - 4 y - 4 = 0$

$x^{2} + y^{2} - 2 x y - 4 x - 4 y - 4 = 0$

$x^{2} + y^{2} + 2 x y - 4 x + 4 y - 4 = 0$

$x^{2} + y^{2} + 2 x y - 4 x - 4 y + 4 = 0$

answer is C.

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Detailed Solution

Tangent at the vertex is $x - y + 1 = 0 - - - - - (1)$

Therefore, the equation of the axis of the parabola is $X + Y = 0 - - - - - - (2)$

$Now, solving (1) and (2), we get A \equiv (- 1 / 2, 1 / 2) .$

$Therefore, Z is (- 1, 1) . (∵ A is midpoint of O Z)$

$Now, the directrix is X-Y+K=0$
$But this passes through Z (- 1, 1) . Therefore, K=0$

$So, the directrix is x - y + 2 = 0$

Therefore, by definition, the equation of the parabola is
given by

$\begin{array}{l} O P = P M \\ or O P^{2} = P M^{2} \\ {(\frac{x - y + 2}{\sqrt{2}})}^{2} = x^{2} + y^{2} \\ or (x - y + 2)^{2} = 2 x^{2} + 2 y^{2} \\ or x^{2} + y^{2} + 4 - 2 x y + 4 x - 4 y = 2 x^{2} + 2 y^{2} \\ or x^{2} + y^{2} + 2 x y - 4 x + 4 y - 4 = 0 \end{array}$