question_answer

Consider four circles\[{{(x\pm 1)}^{2}}+{{(y\pm 1)}^{2}}=1\], then the equation of smaller circle touching these four circle is

A) \[{{x}^{2}}+{{y}^{2}}=3-\sqrt{2}\]

B) \[{{x}^{2}}+{{y}^{2}}=6-3\sqrt{2}\]

C) \[{{x}^{2}}+{{y}^{2}}=5-2\sqrt{2}\]

D) \[{{x}^{2}}+{{y}^{2}}=3-2\sqrt{2}\]

Correct Answer: D

Solution :

\[{{A}_{1}}{{B}_{1}}=\sqrt{4+4}=2\sqrt{2}\]                 \[AB=2\sqrt{2}-2=2(\sqrt{2}-1)\] \[\Rightarrow \]\[OA=AB=\frac{AB}{2}=\sqrt{2}-1\]which is the radius of required circle. Thus, equation of required circle is                 \[{{x}^{2}}+{{y}^{2}}={{(\sqrt{2}-1)}^{2}}=3-2\sqrt{2}\]