# How do you know if the conic section x^2 -9y^2 +36y -45= 0 is a parabola, an ellipse, a hyperbola, or a circle?

Nov 12, 2015

Look at the coefficients on ${x}^{2} \mathmr{and} {y}^{2}$

#### Explanation:

The coefficient on ${x}^{2} = 1$ and the coefficient on ${y}^{2} = - 9$.

If the coefficients are different and also different signs (+ or -), then it must be hyperbola.

Detailed Explanation:

EVERY conic section can be written with the general equation :

$A {x}^{2} + B {y}^{2} + C x + D y + E = 0$

Here is how you distinguish the various conic sections from the coefficients in the general equation:

circle : $A = B$

ellipse : $A \ne B$ but A and B both have the SAME sign (+ or -)

hyperbola : $A \ne B$ but A and B both have DIFFERENT signs

parabola : either A or B equals 0 (only one squared term in the equation)

hope that helped