Question

Consider four circles $(x\pm 1{)}^2+(y\pm 1{)}^2=1$ equation of smaller circle touching these four circles is:

Answer 1

$(x+1{)}^2+(y+1{)}^2=1$

Centre $=(-1,-1)$

$\Rightarrow \left(\right(x-1{)}^2+(y+1{)}^2=1$

Centre $=(1,-1)$

$\Rightarrow (x-1{)}^2+(y-1{)}^2=1$

Centre $=(1,1)$

$\Rightarrow (x+1{)}^2+(y-1{)}^2=1$

Centre $=(-1,1)$

$A^{\prime }{B}^{\prime }=\sqrt{(-1-1{)}^2+(-1-1{)}^2}=\sqrt{4+4}=2\sqrt{2}$

$AB=2\sqrt{2}-1-1=2(\sqrt{2}-1)$

Small radius $=\frac{AB}{2}=\frac{2(\sqrt{2}-1)}{2}=\sqrt{2}-1$ [It passes through origin]

$\therefore$ Equation of small circle $\Rightarrow x^2+y^2=(\sqrt{2}-1{)}^2$.