Gọi đường tròn có tâm \(I\left(a;b\right)\)
Do I tiếp xúc 2 trục tọa độ \(\Rightarrow R=\left|a\right|=\left|b\right|=IM\Rightarrow\left\{{}\begin{matrix}a^2=IM^2\\a^2=b^2\end{matrix}\right.\)
\(\overrightarrow{MI}=\left(a-2;b-1\right)\Rightarrow IM^2=\left(a-2\right)^2+\left(b-1\right)^2\)
\(\Rightarrow\left\{{}\begin{matrix}a^2=\left(a-2\right)^2+\left(b-1\right)^2\\a^2=b^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}b^2-4a-2b+5=0\\\left[{}\begin{matrix}a=b\\a=-b\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}a=b=1\\a=b=5\end{matrix}\right.\)
Có 2 đường tròn:
\(\left[{}\begin{matrix}\left(x-1\right)^2+\left(y-1\right)^2=1\\\left(x-5\right)^2+\left(y-5\right)^2=25\end{matrix}\right.\)
