find m so that this becomes the equation of an ellipse

Question

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.

Hi, welcome to math.SE! Here's a reference and tutorial for typesetting math on this site. — joriki, Apr 03 '16 at 14:35
Do you mean all such $m$? It would seem pretty easy to find just one such $m$, as it just has to be large enough. — joriki, Apr 03 '16 at 14:36
yes every possible m, and they have already excluded 1 and 2 — byrek, Apr 03 '16 at 14:42

score 0 · Answer 1 · answered Apr 03 '16 at 14:45

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 $m\gt2$.

find m so that this becomes the equation of an ellipse

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