question_answer

From the point \[P(-1,-2),\] PQ and PR are the tangents drawn to the circle \[{{x}^{2}}+{{y}^{2}}-6x-8y=0.\]. Then angle subtended by QR at the centre of the circle is

A) \[\pi =2{{\sin }^{-1}}\left( \frac{5}{2\sqrt{13}} \right)\]  

B) \[\pi ={{\sin }^{-1}}\left( \frac{5}{2\sqrt{13}} \right)\]

C) \[{{\cos }^{-1}}\left( \frac{5}{2\sqrt{13}} \right)\]          

D) \[{{\cos }^{-1}}\left( \frac{5}{\sqrt{13}} \right)\]

Correct Answer: A

Solution :

[a] Centre \[(3,4),\] \[CP=\sqrt{52}=2\sqrt{13}\] Angle between tangents \[=2{{\sin }^{-1}}\left( \frac{5}{2\sqrt{13}} \right)\] Angle subtended by QR at \[C=\pi -2{{\sin }^{-1}}\left( \frac{5}{2\sqrt{13}} \right)\]