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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. - Mathematics

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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.

Sum
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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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Chapter 11: Conic Sections - Exercise [Page 203]

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NCERT Exemplar Mathematics [English] Class 11
Chapter 11 Conic Sections
Exercise | Q 17 | Page 203

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