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  • A variable chord PQ,  <img alt="\mathrm{x\cos \theta+y \sin \theta=P}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx%5Ccos%20%5Ctheta&plus;y%20%5Csin%20%5Ctheta%3DP%7D

A variable chord PQ,  \mathrm{x\cos \theta+y \sin \theta=P} of the hyperbola \mathrm{\frac{x^2}{a^2}-\frac{y^2}{2 a^2}=1}, subtends a right angle at the origin. This chord will always touch a circle whose radius is

Option: 1

\mathrm{a}


Option: 2

\mathrm{a \sqrt{2}}


Option: 3

\mathrm{a \sqrt{2}}


Option: 4

\mathrm{2 \mathrm{a} \sqrt{2}}


Answers (1)

best_answer

Combined equation of lines joining the points of intersection of the chord with origin is

\mathrm{\begin{aligned} & \frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{2 \mathrm{a}^2}=\left(\frac{\mathrm{x} \cos \theta+\mathrm{y} \sin \theta}{\mathrm{p}}\right)^2 \\ & \Rightarrow \mathrm{x}^2\left(\frac{\mathrm{p}^2}{\mathrm{a}^2}-\cos ^2 \theta\right)-\mathrm{y}^2\left(\frac{\mathrm{p}^2}{2 \mathrm{a}^2}+\sin ^2 \theta\right)-\mathrm{xy} \sin 2 \theta=0 \end{aligned}}

these lines are mutually perpendicular if 

\mathrm{\begin{aligned} & \frac{p^2}{a^2}-\cos ^2 \theta-\frac{p^2}{2 a^2}-\sin ^2 \theta=0 \\ & \Rightarrow p^2=2 a^2 \Rightarrow p=a \sqrt{2} \end{aligned}}

thus chord becomes \mathrm{x \cos \theta+y \sin \theta=a \sqrt{2}} which is clearly a tangent to the circle having centre at origin and radius is equal to  \mathrm{a \sqrt{2}}

Posted by

Irshad Anwar

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