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

Equation of the chord of contact $PQ: T=S_1$

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