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  • The length of the focal chord of the parabola , which makes an angle <img alt=

The length of the focal chord of the parabola \mathrm{y^2=4 a x}, which makes an angle \mathrm{\alpha} with the x-axis is

Option: 1

\mathrm{4 a \sin ^2 \alpha}


Option: 2

\mathrm{4 a \operatorname{cosec}^2 \alpha}


Option: 3

\mathrm{2 a \operatorname{cosec}^2 \alpha}


Option: 4

\mathrm{a \operatorname{cosec}^2 \alpha}


Answers (1)

best_answer

Let A be \mathrm{\left(a t_1^2, 2 a t_1\right)} and B be \mathrm{\left(a t_2^2, 2 a t_2\right)}. Focal chord AB passes through \mathrm{S(a, 0) \Rightarrow t_1 t_2=-1} (standard result)

\mathrm{ \tan \alpha \equiv \text { slope of } A B=\frac{2 a t_1-2 a t_2}{a t_1^2-a t_2^2} }

\mathrm{\begin{aligned} \text { Or } \quad \tan \alpha & =\frac{2}{t_1+t_2} \\\\ \therefore \quad t_1+t_2 & =2 \cot \alpha\, \, \, \, \, \, \, \, ............(2) \\\\ \text { Length } A B & =\sqrt{\left(a t_2^2-a t_1^2\right)^2+\left(2 a t_2-2 a t_1\right)^2} \\\\ & =a\left(t_2-t_1\right) \sqrt{\left(t_2+t_1\right)^2+4} \\\\ & =a\left(t_2-t_1\right) \sqrt{\left(t_2+t_1\right)^2-4(-1)} \\\\ & =a\left(t_2-t_1\right) \sqrt{\left(t_2+t_1\right)^2-4\left(t_1 t_1\right)} \\\\ & =a\left(t_2-t_1\right) \sqrt{\left(t_2-t_1\right)^2} \\\\ \text { Length } A B & =a\left(t_2-t_1\right)^2(\text { standard result }) \\\\ & =a\left[\left(t_2+t_1\right)^2-4 t_1 t_2\right] \\\\ & =a\left[4 \cot ^2 \alpha-4(-1)\right]=4 a \operatorname{cosec}^2 \alpha \end{aligned}}

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shivangi.shekhar

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