# How do you determine circle, parabola, ellipse, or hyperbola from equation 4x^2 - 9y^2 - 16x +18y -11 = 0?

Nov 12, 2015

EVERY conic section can be written with the general equation :

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

#### Explanation:

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)

So, in the problem stated above, since $A \ne B$ ($4 \ne 9$), but A and B have different signs (4 and -9), it is a hypebola !

hope that helped