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.

Answer 1

Given equation of the parabola is $y^2=4ax$

Let the coordinates of any point P on the parabola be $a{t}^2,2at$

In $△POA,tan\theta =\frac{2at}{a{t}^2}=\frac{2}{t}$

$\Rightarrow tan\theta =\frac{2}{t}\Rightarrow t=2cot\theta$

$\therefore$ length of OP=$\sqrt{(0-a{t}^2{)}^2+(0-2at{)}^2}$

=$\sqrt{a^2{t}^2+4a^2{t}^2}$

=$at\sqrt{t^2+4}$

=$2acot\theta \sqrt{4co{t}^2\theta +4}$

=$4acot\theta \sqrt{1+co{t}^2\theta }$

=$4acot\theta .cosec\theta$

=$\frac{4acos\theta }{sin\theta }.\frac{1}{sin\theta }=\frac{4acos\theta }{si{n}^2\theta }$