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Semi transverse axis of hyperbola is 5. Tangent at point P and normal to this tangent meet conjugate axis at A and B, respectively. The circle on AB as diameter passes through tow fixed points, the distance between which is 20. Find the eccentricity of hyperbola.

Answer» <html><body><p></p>Solution :<a href="https://interviewquestions.tuteehub.com/tag/consider-2017521" style="font-weight:bold;" target="_blank" title="Click to know more about CONSIDER">CONSIDER</a> hyperbola `(x^(2))/(a^(2))-(y^(2))/(<a href="https://interviewquestions.tuteehub.com/tag/b-387190" style="font-weight:bold;" target="_blank" title="Click to know more about B">B</a>^(2))=1`. <br/> Tangent to it at any point `P(a sec theta, b tan theta)` is <br/> `(x)/(a) sec theta-(y)/(b) tan theta=1` <br/> It <a href="https://interviewquestions.tuteehub.com/tag/meets-1092960" style="font-weight:bold;" target="_blank" title="Click to know more about MEETS">MEETS</a> y-axis at `A(0,-b cot theta)`. <br/> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/CEN_CG_C07_S01_057_S01.png" width="80%"/> <br/> <a href="https://interviewquestions.tuteehub.com/tag/normal-1123860" style="font-weight:bold;" target="_blank" title="Click to know more about NORMAL">NORMAL</a> at point P is <br/> `ax cos theta+by cot theta=a^(2)e^(2)` <br/> It meets y-axis at `B(0,(a^(2)e^(2))/(b) tan theta)` <br/> Now, circle with diameter as AB is <br/> `x^(2)+(y+b cot theta)(y-(a^(2)e^(2))/(b)tantheta)=0` <br/> `rArr""x^(2)+y^(2)-(a^(2)e^(2))+(b cot theta-(a^(2)e^(2))/(b)tantheta)y=0` <br/> This circle passes through fixed point `(pm ae,0)`, <a href="https://interviewquestions.tuteehub.com/tag/distnace-2587924" style="font-weight:bold;" target="_blank" title="Click to know more about DISTNACE">DISTNACE</a> between which is 2ae. <br/> `2ae=20"(given)"` <br/> `therefore""e=2`</body></html>


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