Question

P is any point on the parabola $y^2=4ax$ whose vertex is A. PA is produced to meet the directrix in D & M is the foot of the perpendicular from P on the directrix. The angle subtended by MD at the focus is:

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{3}$
• C. $\frac{5\pi }{12}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is D $\frac{\pi }{2}$

Let $P(a{t}^2,2at)$ be a point on parabola
Equation of AP
$y-0=\frac{2at}{a{t}^2}(x-0)$
$y=\frac{2}{t}x$
AP intersect directrix $x=-a$ at D
$y=-\frac{2a}{t}$
$\therefore D(-a,-\frac{2a}{t})$
Slope of $DF=\frac{0+\frac{2a}{t}}{a+a}=\frac{1}{t}$
PM is perpendicular to $x=-a$
$\therefore M(-a,2at)$
Slope of $MF=\frac{0-2at}{a+a}=-t$
Slope of DF $\times$ Slope of MF $=\frac{1}{t}\times -t$ $=-1$
$\therefore$ DF and MF are perpendicular
$\therefore \theta ={90}^0$ or $\frac{\pi }{2}$