Question

If the line segment joining the vertex of the parabola $y^{2}=4 a x$ and a point on the parabola, makes an angle $\theta$ with the positive $X$ -axis, then the length of that line segment is

• A. $\frac{4 a \sin \theta}{\cos ^{2} \theta}$
• B. $\frac{4 a \cos \theta}{\sin ^{2} \theta}$
• C. $4 a \sin \theta \cdot \cos ^{2} \theta$
• D. $4 a \cos \theta \cdot \sin ^{2} \theta$

Answer 1

Answer: (B)

Equation of given parabola is $y^{2}=4 a x$ having vertex $V(0,0)$
Let a point on the parabola $P\left(a t^{2}, 2 a t\right)$, so slope of line joining point $V(0,0)$ and $P\left(a t^{2}, 2 a t\right)$ is
$\frac{2 a t-0}{a t^{2}-0}=\tan \theta$ (given)
$\Rightarrow \quad \frac{2}{t}=\tan \theta $
$\Rightarrow t=2 \cot \theta$
Now, length of line segment $V P=\sqrt{\left(a t^{2}\right)^{2}+(2 a t)^{2}}$
$=a \sqrt{t^{4}+4 t^{2}}=a \sqrt{(2 \cot \theta)^{4}+4(2 \cot \theta)^{2}} $
$=4 a \,\cot \,\theta \,\text{cosec} \,\theta=\frac{4 a \,\cos \,\theta}{\sin ^{2} \theta}$