For the equation $$(m − 2)x^2 + (y − 1)^2 − (m − 1)(m − 2) = 0 \textrm{ and } m \in \mathbb{R} \setminus \begin{Bmatrix}1, 2\end{Bmatrix}$$ Find $m$ so that this becomes the equation of an ellipse.
I have honestly no idea how to do this, I understand why $m$ should me different from 1 and 2 but i can't find any specific values.
The equation describes an ellipse if the quadratic terms for $x$ and $y$ have the same sign and the constant term has the opposite sign. (If the constant term has the same sign, the equation has no solutions, and if the quadratic terms have opposite signs, the equation describes a hyperbola.)
These conditions are fulfilled if and only if $m\gt2$.