Question

Find the length of the segment joining the vertex of the parabola $y^2=ax$ and a point on the parabola where the line segment makes an angle ${}^{\prime }{\theta }^{\prime }$ to the x-axis.

• A. $\frac{2a\cos \theta }{{\sin }^2\theta }$
• B. $\frac{4a\cos \theta }{{\sin }^2\theta }$
• C. $\frac{4a\cos \theta }{3{\sin }^2\theta }$
• D. None of the above

Answer 1

The correct option is A $\frac{2a\cos \theta }{{\sin }^2\theta }$

Let any point $(h,k)$ will satisfy.
$y^2=4ax$ i.e., $k^2=4ah....\left(i\right)$
Let $a$ line $OP$ makes an angle $\theta$ from the axis.
$\therefore ln△OAP.\sin \theta =\frac{PA}{OP}$
$\Rightarrow \sin \theta =\frac{K}{I}\Rightarrow k=I\sin \theta$
and $\cos \theta =\frac{OA}{OP}\Rightarrow \cos \theta =\frac{h}{I}$
$\Rightarrow h=I\cos \theta$
From Eq. (i), we get
$I^2{\sin }^2\theta =4a\times I\cos \theta$ (put $k=I\sin \theta ,h=I\cos \theta )$
$\Rightarrow I=\frac{4a\cos \theta }{{\sin }^2\theta }$.