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If (-1,-1) is the focus and $x + y + 4 = 0$ is the directrix of a parabola then its vertex is

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$(- \frac{3}{2}, - \frac{3}{2})$

$(- \frac{5}{2}, - \frac{5}{2})$

$(- \frac{1}{4}, - \frac{1}{4})$

$(\frac{1}{4}, \frac{1}{4})$

answer is A.

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Detailed Solution

$S = (- 1, - 1), L = x + y + 4 = 0 & P (x, y)$

$\frac{S P}{P M} = 1 \Rightarrow {(S P)}^{2} = {(P M)}^{2}$

$x - y = 0 \to (1) & x + y + 3 = 0 \to (2)$

Solve (1) & (2) $\Rightarrow x = \frac{- 3}{2}, y = \frac{- 3}{2}$

Vertex is $(\frac{- 3}{2}, \frac{- 3}{2})$

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