question_answer

The angle between the tangents from \[(\alpha ,\beta )\]to the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\], is

A)            \[{{\tan }^{-1}}\left( \frac{a}{\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}-{{a}^{2}}}} \right)\]  

B)            \[{{\tan }^{-1}}\left( \frac{\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}-{{a}^{2}}}}{a} \right)\]

C)            \[2{{\tan }^{-1}}\left( \frac{a}{\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}-{{a}^{2}}}} \right)\]               

D)            None of these

Correct Answer: C

Solution :

           \[\tan \frac{\theta }{2}=\frac{C{{T}_{1}}}{P{{T}_{1}}}=\frac{a}{\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}-{{a}^{2}}}}\]                    \[\frac{\theta }{2}={{\tan }^{-1}}\frac{a}{\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}-{{a}^{2}}}}\Rightarrow \theta =2{{\tan }^{-1}}\frac{a}{\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}-{{a}^{2}}}}\].