Question

Let the curve C be the mirror image of the parabola $y^2=4x$ with respect of the line $x+y+4=0$. If A and B are the points of intersection of C with the line $y=-5$, then the distance between A and B is

Answer 1

Answer: 4

Let any point on the parabola $y^2=4x$ is $(t^2,2t)$
Let its mirror image with respect to the line $x+y+4=0$ is (h, k) then
$\frac{h-t^2}{1}=\frac{k-2t}{1}=\frac{-2\left(t^2+2t+4\right)}{2}$
$\therefore h=-2t-4,k=-t^2-4$
So $k+4=-{\left(\frac{h+4}{2}\right)}^2$
$\Rightarrow {\left(h+4\right)}^2=-4\left(k+4\right)$
So locus of C is
${\left(x+4\right)}^2=-4\left(y+4\right)$
It intersects $y=-5$
So ${\left(x+4\right)}^2=4$
$\Rightarrow x+4=\pm 2$
$\Rightarrow x=-2,-6$
$\therefore |x_1-x_2|=4$