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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.
Solution
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)`.
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