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

  $x^2+y^2=5-2 \sqrt{2}$
• Correct answer
  D

  $x^2+y^2=3-2 \sqrt{2}$

Solution

$A_1 B_1=\sqrt{4+4}=2 \sqrt{2}$
$A B=2 \sqrt{2}-2=2(\sqrt{2}-1)$ $\Rightarrow O A=O B=\frac{A B}{2}=\sqrt{2}-1$ which is the radius of required circle. Thus, equation of required circle is $x^2+y^2=(\sqrt{2}-1)^2=3-2 \sqrt{2}$

Hence, the correct answer is option: 'D'.