how do i tell if an equation is a circle, parabola, an ellipse, or hyperbola?
an example is 5y^2-9y-12x^2-3x+6=0
what do i look at to determine its geometry?
Since the leading factors of the squares are different (value and sign) and there aren't any summands containing $\displaystyle x \cdot y$ , this equation describes a hyperbola.
If the axes of the conic section are parallel to the coordinate axes, you can determine the kind of conic by:
1. The leading factors of the squares are both positive and equal: It's a circle.
2. The leading factors of the squares are positive but unequal: It's an ellipse.
3. The leading factors of the squares are different in value and sign: It's a hyperbola.
Indeed: http://www.wolframalpha.com/input/?i...1y%5E2+%3D+375.
That's what I get for taking things on face value and not checking ....