Let A and B be the two points of intersection of the line $y+5=0$ and the mirror image of the parabola $y^2=4 x$ with respect to the line $x+y+4=0$. If d denotes the distance between A and B , and a denotes the area of $ riangle S A B$, where $S$ is the focus of the parabola $y^2=4 x$, then the…

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Let A and B be the two points of intersection of the line $y+5=0$ and the mirror image of the parabola $y^2=4 x$ with respect to the line $x+y+4=0$. If d denotes the distance between A and B , and a denotes the area of $	riangle S A B$, where $S$ is the focus of the parabola $y^2=4 x$, then the value of $(a+d)$ is ________ -

Aicharya · Accepted Answer

To find image of $P\left(t^2, 2 tight)$ $\begin{aligned} & \frac{x-t^2}{1}=\frac{y-2 t}{1}=\frac{-2\left(t^2+2 t+4ight)}{1^2+1^2}=-(t+1)^2-3 \ & x=t^2-(t+1)^2-3=-2 t-4 \ & y=2 t-(t+1)^2-3=-t^2-4 \ & t=\frac{-x-4}{2} \ & \Rightarrow y+4=-\left(\frac{-x-4}{2}ight)^2 \ & \Rightarrow(y+4)=-\frac{(x+4)^2}{4} \ & \Rightarrow x^2=-4 y \ & \Rightarrow 	ext { Focus }(-4,-5) \end{aligned}$ $	ext { Also, } y=-5 	ext { intersect }$ $\begin{aligned} & 	herefore(-4)(-1)=(x+4)^2 \ & 4=(x+4)^2 \ & x+4= \pm 2 \ & x=-2,-6 \ & \Rightarrow d=4 \ & a=\frac{1}{2}\left|\begin{array}{ccc}1 & 0 & 1 \ -2 & -5 & 1 \ -6 & -5 & 1\end{array}ight| \ & =\frac{1}{2}|[1(-5+5)+1(10-30)]| \ & =\frac{1}{2}(20) \ & a=10 \ & 	herefore \quad a+d=14\end{aligned}$

Let A and B be the two points of intersection of the line $y+5=0$ and the mirror image of the parabola…

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