Question

Find the length of the line segment joining the vertex of the parabola y² = 4ax and a point on the parabola where the line segment makes an angle θ to the x-axis.

Answer 1

Equation of parabola is y² = 4ax

Let P(at², 2at) be any point on the parabola.

In ΔPOA, we have

tan θ = `(2at)/(at^2) = 2/t`

⇒ t = `2/(tan theta)`

⇒ t = 2 cot θ  .....(i)

OP = `sqrt((at^2 - 0)^2 + (2at - 0)^2)`

= `sqrt(a^2t^4 + 4a^2t^2)`

= `at sqrt(t^2 + 4)`

= `a xx 2 cot theta sqrt(4 cot^2 theta + 4)`  .....[∵ t = 2 cot θ]

= `2a cot theta . 2sqrt(cot^2theta + 1)`

= 4a cot θ.cosec θ

= `4a * costheta/sintheta * 1/sintheta`

= `(4a costheta)/(sin^2theta)`

Hence, the required length = `(4a costheta)/(sin^2theta)`.