Find the length of the line segment joining the vertex of the parabola `y^2=4a x` and a point on the parabola where the line segment make and angle `theta` to the `x`- axis.

Explanation

Given equation of the parabola is `y^(2)`=4ax

Let the coordinates of any point P on the parabola be `(at^(2),2at)`.
In `DeltaPOA` `tantheta=(2at)/(at^(2))=2/t`
`rArr tantheta=2/trArrt=2cottheta`
`therefore` length of OP=`sqrt((0-at^(2))^(2)+(0-2at)^(2))`
`=sqrt(a^(2)t^(4)+4a^(2)t^(2))`
`=atsqrt(t^(2)+4)`
`=2acotthetasqrt(4cot^(2)theta+4)`
`=4acotthetasqrt(1+cot^(2)theta)`
`4acotthetacdotcosectheta`
`=(4acostheta)/(sintheta)cdot1/(sintheta)=(4acostheta)/(sin^(2)theta)`