Suppose $f(x,y)=0$ is the equation of the circle such that $f(x,1)=0$ has equal roots (each equal to $2$) and $f(1,x)=0$ also has equal roots (each equal to $0$). Find the equation of the circle.
My attempt: I could only gather that the circle meets the lines $x=1$ and $y=1$ at $(1,0)$ and $(2,1)$ respectively.
The condition that $f(1,y)=0$ has a double root at $y=0$ means that the line $x=1$ has one point intersection with the circle at $(1,0)$, which is possible only when the line is tangent and $(1,0)$ is the tangent point. Similarly for the second condition. Now, can you think of a circle that touches two lines at the given points?
Here is the spoiler: