question_answer

A circle passes through (0, 0) and (1, 0) and touches the circle \[{{x}^{2}}+{{y}^{2}}=9\], then the centre of circle is [IIT 1992]

A)            \[\left( \frac{3}{2},\frac{1}{2} \right)\]                                     

B)            \[\left( \frac{1}{2},\frac{3}{2} \right)\]

C)            \[\left( \frac{1}{2},\frac{1}{2} \right)\]                                     

D)            \[\left( \frac{1}{2},\ \pm \sqrt{2} \right)\]

Correct Answer: D

Solution :

           Radius of the circle \[r=\frac{3}{2}\]                    \[\frac{1}{4}+{{k}^{2}}=\frac{9}{4}\Rightarrow k=\pm \sqrt{2}\]                    Hence centre is\[\left( \frac{1}{2},\ \pm \sqrt{2} \right)\].