Question

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

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

Answer 1

Answer: (B)

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