Question

If three normals are drawn from the point $(6,0)$ to the parabola $y^2=4ax$ of which two are mutually perpendicular, then length of its latus rectum is

Answer 1

Answer: 8

Normal of parabola $y^2=4ax$ is
$y=mx-a{m}^3-2am$ and it is passing through $(6,0)$
$\Rightarrow 6m-a{m}^3-2am=0$
$\Rightarrow m=0,$ or, $6-2a-a{m}^2=0$
$m=0\Rightarrow$ normal is $x$ axis
$a{m}^2+2a-6=0$
$\Rightarrow \frac{2a-6}{a}=-1\left(\because m_1{m}_2=-1\right)$
We cannot draw a normal parallel to axis of parabola.
$2a-6=0-a\Rightarrow 3a=6\Rightarrow a=2$
$\Rightarrow$ Length of latus rectum $=8$