Let pirectrix latus rectum$=4b$

equation of parabola,

$(y-0)^2 =4b (x-a+b)$

$2y \large\frac{dy}{dx} $$=4b$

$y \large\frac{dy}{dx} $$=2b$

$y= 2 \bigg(\large\frac{dy}{dx} \bigg) \bigg( x- a +\large\frac{dy}{dx}\bigg)$

$y= 2\large\frac{ xdy}{dx} $$-2a \large\frac{dy}{dx} $$+2 \bigg(\large\frac{dy}{dx}\bigg)^2$

$\bigg(\large\frac{dy}{dx}\bigg)= \frac{y}{2}$

Hence c is the correct answer.