question_answer

The locus of the point of intersection of the tangents at the extremities of a chord of the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]which  touches the circle \[{{x}^{2}}+{{y}^{2}}=2ax\]is

A)            \[{{y}^{2}}=a\text{ }(a-2x)\]     

B)            \[{{x}^{2}}=a\text{ }(a-2y)\]

C)            \[{{x}^{2}}+{{y}^{2}}={{(y-a)}^{2}}\]                       

D)            None of these

Correct Answer: A

Solution :

           \[T\equiv hx+ky-{{a}^{2}}=0\]                                \[\Rightarrow a=\frac{ah+0-{{a}^{2}}}{\sqrt{{{h}^{2}}+{{k}^{2}}}}\]                    \[P(3,\,4)\].                    Þ \[{{k}^{2}}=a(a-2h)\]                    \ The locus is\[{{y}^{2}}=a(a-2x)\].