My problems #8

Clearly the pair of lines formed by $xy = 0$ are $x$ axis and $y$ axis.

The second pair of lines is:- $xy+1=x+y$ $xy-x-y+1=0$ $(x-1)(y-1) =0$ Hence the pair of lines by this equation is $x=1$ and $y=1$ .

Hence the square formed has side equal to $1 unit$ .

Let radius of circle with center $C_{1}$ is $r$ . Hence radius of circle with Center $C_{2}$ is $2r$ .

From symmetry of square we can claim that $C_{1}$ and $C_{2}$ lie on the diagonal of the square. Looking at this diagonal we can the following equation:- $\sqrt{2}r+r+2r+2\sqrt{2}r = \sqrt{2}$ $3r+3\sqrt{2} = \sqrt{2}$ $r=\dfrac{2-\sqrt{2}}{3}$