affine transformation proof

[Poirot]

prove that there is no affine transformation mapping hyperbola $x^2-y^2=1$ to unit circle $x^2+y^2=1$.

 

[Turgul]

Are you trying to show this over the real numbers? If so, in $\mathbb{R}^2$, affine transformations either turn $\mathbb{R}^2$ into a point or a line, or they are bijective (homeomorphisms!)(why are these the only possibilities?).

Since the circle does not lie in a point nor a line, if such an affine transformation did exist, it would thus have to be a homeomorphism of $\mathbb{R}^2$ to itself. On the other hand, over $\mathbb{R}$, the hyperbola is not connected, but the circle is. Why is this bad?