# How do you know if 5x^2-3y^2-20x-36y=103 is a hyperbola, parabola, circle or ellipse?

Oct 24, 2015

It has coefficients for both ${x}^{2}$ and ${y}^{2}$ of different signs and no $x y$ term, so it is a hyperbola.

#### Explanation:

Without an $x y$ term:

A circle would have equal coefficients for ${x}^{2}$ and ${y}^{2}$

An ellipse would have coefficients of the same sign for ${x}^{2}$ and ${y}^{2}$

A parabola would have either an ${x}^{2}$ term or a ${y}^{2}$ term but not both.

$5 {x}^{2} - 3 {y}^{2} - 20 x - 36 y = 103$

graph{5x^2-3y^2-20x-36y =103 [-39.58, 40.42, -25.32, 14.68]}

If these conic sections are rotated then they can have an $x y$ term and it can be more difficult to tell them apart.