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  • A circle is drawn considering any focal chord   of the parabola <img alt="\mathrm{y^2=

A circle is drawn considering any focal chord \mathrm{P Q}  of the parabola \mathrm{y^2=4 a x}  as diameter which intersects the tangent at \mathrm{P^{\left(a t^2, 2 a t\right)}} in A. Then the length of the side \mathrm{A Q } of \mathrm{\triangle A S Q }  is (where S is the focus of the parabola)
 

Option: 1

\mathrm{\frac{a}{t^2}\left(1+t^2\right)^{3 / 2}}


Option: 2

\mathrm{ a t^2\left(1+t^2\right)^{3 / 2} }


Option: 3

\mathrm{ \frac{a t^2}{\left(1+t^2\right)^{3 / 2}}}


Option: 4

 none of these
 


Answers (1)

best_answer

Since required distance is perpendicular distance of Q from tangent at P.

Equation of tangent at \mathrm{P \ is \ x-t y+a t^2=0 \quad\left[Q:\left(\frac{a}{t^2},-\frac{2 a}{t}\right)\right]}

Required distance = \mathrm{\left|\frac{\frac{a}{t^2}-t \times\left(-\frac{2 a}{t}\right)+a t^2}{\sqrt{1+t^2}}\right|=t^{\frac{a}{t^2} \frac{\left(1+t^2\right)^2}{\sqrt{\left(1+t^2\right)}}}=\frac{a}{t^2}\left(1+t^2\right)^{3 / 2}}

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shivangi.bhatnagar

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