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

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
$\frac{h-t^{2}}{1}=\frac{k-2t}{1}=\frac{-2\left(t^{2}+2t+4\right)}{2}$
$∴ h = -2t - 4, k = - t^{2} - 4$
So $k + 4 = - \left(\frac{h+4}{2}\right)^{2}$
$⇒ \left(h + 4\right)^{2} = - 4\left(k + 4\right)$
So locus of C is
$\left(x + 4\right)^{2} = - 4\left(y + 4\right)$
It intersects $y = -5$
So $\left(x + 4\right)^{2} = 4$
$⇒ x + 4 = \pm \,2$
$⇒ x = -2, -6$
$∴ |x_{1} - x_{2}| = 4$