Question

Find the length of the line segment joining the vertex of the parabola $y^2=4ax$ and a point on the parabola, where the line segment makes an angle $\theta$ 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 these

Answer 1

Answer: (B)

$\frac{4a\cos \theta }{{\sin }^2\theta }$

Let any point $(h,k)$ will satisfy
$y^2=4ax$ i.e., $k^2=4ah$ ....(i)
Let a line $OP$ makes an angle $\theta$ from the $X$-axis.
$\therefore$ In $\Delta OAP$,
$\sin \theta =\frac{PA}{OP}$
$\Rightarrow \sin \theta =\frac{k}{l}$
$\Rightarrow k=l\sin \theta$
and $\cos \theta =\frac{OA}{OP}$
$\Rightarrow \cos \theta =\frac{h}{l}\Rightarrow h=l\cos \theta$
Hence, from equation (i), we get
$l^2{\sin }^2\theta =4a\times l\cos \theta$ $(putk=l\sin \theta ,h=l\cos \theta )$
$\Rightarrow l=\frac{4a\cos \theta }{{\sin }^2\theta }$.