To which parameter C does the following equation:
$$(2-c)x^2+(3-c)y^2+2x+8y+5=0$$
is an equation for:
1) Ellipse OR Circle
2) Hyperbole
3) Parabola
Well, as far as I know that in order to get a circle we must have the condition $a = b \ne 0 $ and here I will call $(2-c)=a$ and $(3-c)=b$, and this can't happen for any C, so I said that Option one can't get a Circle so I wanted to check if we can get an Ellipse, (because it includes the OR option, we might find a C which makes it true), so to get and Ellipse we must have the condition $a \ne b \ne 0$ and that can happen with $c=4$ for example (and this case we want "a" and "b" with an equal sign). and to get Hyperbole last condition must be fulfilled but "a" and "b" must be with different signs, so that can't happen and I said that this option is false. and about last one, to get Parapola $C=3$.
whats wrong with my solution?
NOTE: there is one mistake between the things that I have said.
The discussion only rests on the sign of the discriminant, $-(2-c)(3-c)$.
When negative, there are no asymptotes, hence ellipse (or circle).
When positive, two asymptotes, hyperbola.