Question

The length of the line segment joining the vertex of the parabola $y ^{2}=4 ax$ and a point on the parabola where the line segment makes an angle $\theta$ to the $x$ -axis is $\frac{4 am }{ n } .$ Here, $m$ and $n$ respectively are

• A. $\sin \theta, \cos \theta$
• B. $\cos \theta, \sin \theta$
• C. $\cos \theta, \sin ^{2} \theta$
• D. $\sin ^{2} \theta, \cos \theta$

Answer 1

Answer: (C)

Let any point $P ( h , k )$ will satisfy $y^{2}=4 a x$ i.e, $k^{2}=4 a h$
Let a line $ OP$ makes an angle $\theta$ from the $x$ -axis.
$\therefore \text { In } \Delta OAP , \sin \theta=\frac{ PA }{ OP } $
$ \sin \theta=\frac{ k }{l} \Rightarrow k =l \sin \theta $
and $ \cos \theta=\frac{ OA }{ OP } \Rightarrow \cos \theta=\frac{ h }{l} \Rightarrow h=\ell \cos \theta$

Hence, from eq.(i), we get
$l^{2} \sin ^{2} \theta=4 a \times l \cos \theta \,\,\,\, ($ put $k =l \sin \theta, h =l \cos \theta)$
$\Rightarrow l=\frac{4 a \cos \theta}{\sin ^{2} \theta}$