question_answer

If a circle of radius 'r' is concentric with ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1,\], then the common tangent is inclined to major axis at an angle

A)  \[{{\tan }^{-1}}\,\sqrt{\frac{{{r}^{2}}-{{b}^{2}}}{{{a}^{2}}-{{r}^{2}}}}\]    

B)  \[{{\tan }^{-1}}\,\sqrt{\frac{{{r}^{2}}-{{a}^{2}}}{{{b}^{2}}-{{r}^{2}}}}\]

C)  \[{{\tan }^{-1}}\,\sqrt{\frac{{{r}^{2}}-{{b}^{2}}}{{{r}^{2}}-{{a}^{2}}}}\]    

D)  \[{{\tan }^{-1}}\,\sqrt{\frac{{{r}^{2}}-{{a}^{2}}}{{{r}^{2}}-{{b}^{2}}}}\]

Correct Answer: A

Solution :

 Let equation of circle is \[t=0\]. Tangent to ellipse is \[u=-6\,\,m{{s}^{-1}}\] If it is a tangent to the circle, then it is perpendicular form (0,0) is equal to radius. \[t=6\]  \[v=6\,m{{s}^{-1}}\] \[W=\frac{1}{2}m{{v}^{2}}-\frac{1}{2}m{{u}^{2}}\Rightarrow \,W=0\]     \[{{h}_{A}}<{{h}_{B}}\] \[{{d}_{A}}>{{d}_{B}}\]  \[{{c}_{2}}\]