$C_1$ is a circle of radius ‘1’ touching the x - axis and the y - axis in first quadrant. $C_2$ is another circle of radius >1 and touching the axes as well as the circle $C_1$. Then the radius of $C_2$ is

$$\begin{gathered} Both circles touches axis, i.e. \mid h\mid =\mid k\mid =r=1 \\ \Rightarrow g^2=f^2=c, and r=\sqrt{g^2+f^2-c} \\ Given, r=1 \\ \Rightarrow g^2+f^2-c =1 \\ \Rightarrow g=f=c \\ Equation of circle C_1 is {(x+1)}^2+{(y+1)}^2=1 \\ \Rightarrow x^2+y^2+2x+2y+1=0 \\ Circle C_2 also touches both axes g^2=f^2=c \\ r=\sqrt{g^2+f^2-c } \\ \Rightarrow r=g=f \\ Also, C_2 touches circle C_1 \\ r_1+r_2=\sqrt{{(g-1)}^2+{(f-1)}^2 } \\ \Rightarrow 1+r=\sqrt{{(g-1)}^2+{(f-1)}^2 } \\ {(1+g)}^2={(g-1)}^2+{(f-1)}^2 Since, g=f \\ We get, g^2-6g+1=0 \\ \Rightarrow g=3\pm 2\sqrt{2} =r=f \\ Since, r>1,r=3+2\sqrt{2} \end{gathered}$$