Question

If $P$ be a point on the parabola $y^2=4ax$ with focus $F$. Let $Q$ denote the foot of the perpendicular from $P$ onto the diretrix. 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 $1$

Equation of parabola is
$y^2=4ax.....\left(i\right)$
Let the parametric coordinate of point $P$ on the parabola is $(a,2a)$
Now, $QF=2\sqrt{2}a$.
$PQ=2a$ and $PF=2a$
We observe that ${QF}^2={PQ}^2+{PF}^2$
$\Rightarrow$ $8a^2=4a^2+4a^2=8a^2$
So, $△QPF$ form a right angle isoceles triangle in which $\angle PQF=\angle PFQ$
$\Rightarrow$ $\tan \angle PQF=\tan \angle PFQ$
$\Rightarrow$ $\frac{\tan \angle PQF}{\tan \angle PFQ}=1$