Question

Let P be a point on the parabola $y^2=4ax$ with focus F. Let Q denote the foot of the perpendicular from P onto the directrix. Then $\frac{tan\angle PQF}{tan\angle PFQ}$ is

• A. $1$
• B. $\frac{1}{2}$
• C. $2$
• D. $\frac{1}{4}$

Answer 1

The correct option is A $1$

Let P be the point on parabola $y^2=4ax$ With focus F .$Let Q denote foot of perpendicular from P onto directrix

$\frac{\tan PQF}{\tan PFQ}$ is:

$m_{PQ}=0m_{PF}=\frac{2at}{a{t}^2-a}=\frac{2t}{t^2-1}{m}_{QF}=\frac{2at}{-a-a}=-t\tan \angle PQF=\frac{m_{PQ}-m_{PF}}{1+\left(m_{PQ}\right)\left(m_{PF}\right)}=t\tan \angle PFQ=\frac{m_{PQ}-m_{QF}}{1+\left(m_{PQ}\right)\left(m_{QF}\right)}=t\frac{\tan PQF}{\tan PFQ}=\frac{t}{t}=1$