Question

Tangents are drawn from any point on the directrix of $y^{2}=16x$ to the parabola. If the locus of the midpoint of chords of contact is a parabola, then its length (in units) of the latus rectum is

Answer 1

Equation of the chord of contact $PQ:T=S_{1}$
$\Rightarrow yk-8\left(x + h\right)=k^{2}-16h\ldots \ldots .\left(i\right)$
According to the property, the chord of contact w.r.t. any point on the directrix always passes through focus $\left(4,0\right)$ i.e., $\left(4,0\right)$ satisfy the equation $\left(i\right)$
$\Rightarrow -8\left(4 + h\right)=k^{2}-16h$
$\Rightarrow -32-8x=y^{2}-16x$
$y^{2}=8\left(x - 4\right)$
$\Rightarrow $ Length of the latus rectum is $8$
units