If (-1,-1) is the focus and x+y+4=0 is the di | vedclass

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(A) The focus $S$ is $(-1, -1)$ and the directrix is $x+y+4=0$.The vertex is the midpoint of the focus and the point of intersection of the axis of th

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

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(A) The focus $S$ is $(-1, -1)$ and the directrix is $x+y+4=0$.The vertex is the midpoint of the focus and the point of intersection of the axis of the parabola and the directrix.The axis of the parabola is perpendicular to the directrix and passes through the focus. Since the directrix is $x+y+4=0$ (slope $-1$),the axis has slope $1$ and passes through $(-1, -1)$.The equation of the axis is $y - (-1) = 1(x - (-1)) \Rightarrow y = x$.The intersection of the axis $y=x$ and the directrix $x+y+4=0$ is $x+x+4=0$ $\Rightarrow 2x = -4$ $\Rightarrow x = -2$. Thus,$y = -2$.The point of intersection is $Z(-2, -2)$.The vertex is the midpoint of $S(-1, -1)$ and $Z(-2, -2)$.Vertex $= \left(\frac{-1-2}{2}, \frac{-1-2}{2}\right) = \left(-\frac{3}{2}, -\frac{3}{2}\right)$.

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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