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  • The ratio of the length of a focal chord of a parabola and having inclination with the axis of parabola to its la

The ratio of the length of a focal chord of a parabola and having inclination \alpha with the axis of parabola to its latus rectum is

Option: 1

\operatorname{cosec}^{2} \alpha


Option: 2

\sec ^{2} \alpha


Option: 3

\sin ^{2} \alpha


Option: 4

\tan ^{2} \alpha


Answers (1)

best_answer

Taking the equation of parabola \mathrm{y^{2}=4 a x}

Equation of focal chord \mathrm{y=(x-a) \tan \alpha \, If \, \left(\mathrm{x}_{1}, \mathrm{y}_{1}\right) \, and \, \left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)} are the extremities of the focal chord and its length be \mathrm{L}, the

\mathrm{\mathrm{L}^{2}=\left(\mathrm{x}_{2}-\mathrm{x}_{1}\right)^{2}+\left(\mathrm{y}_{2}-\mathrm{y}_{1}\right)^{2}=\left(\mathrm{x}_{2}-\mathrm{x}_{1}\right)^{2} \sec ^{2} \alpha x_{1}, x_{2}}   are roots of \mathrm{x^{2}-2 a\left(1+2 \cot ^{2} \alpha\right) x+a^{2}=0}

and \mathrm{\mathrm{L}^{2}=\left[4 \mathrm{a}^{2}\left(1+2 \cot ^{2} \alpha\right)-4 \mathrm{a}^{2}\right] \sec ^{2} \alpha=16 \mathrm{a}^{2} \operatorname{cosec}^{4} \alpha}.

\mathrm{\therefore}   required ratio \mathrm{=\operatorname{cosec} y: 1}.

Posted by

manish painkra

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