Q. The length of the chord of the parabola $y^{2} = 4ax$, which passes through the vertex and makes an angle $\alpha$ with the axis of the parabola is
A) $4a\ \cos\ \alpha\ cosec^{2}\alpha$
B) $4a\ \cos\ \alpha\ cosec^{2}\alpha$
C) $a\ \cos\ \alpha\ cosec^{2}\alpha$
D) $a\ cos^{2}\alpha\ a\ cosec\ \alpha$
Explanation
(b)
Let $A$ be the vertex of the parabola and $AP$ is chord of parabola such that slope of $AP$ is $\cot\alpha$. Let coordinates of $P$ be $\left( t^{2},\ 2t \right)$, which is a point on the parabola.
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Slope of $AP = \frac{2t}{t^{2}}$
$\Rightarrow \ \tan{\alpha = \frac{2}{t}}$
$t = 2{cost}\alpha$
In $\mathrm{\Delta}\ APB,\ AP = \sqrt{4t^{2} + t^{4}}$
$= t\sqrt{4 + t^{2}}$
$= 2\cot{\alpha\sqrt{4(1 + \cot^{2}{\alpha)}}}$
$= 4\cot\alpha\ cosec\ \alpha$
$= 4\cos\alpha cosec^{2}\alpha$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.