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

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Answer 1 · 2015-10-24T15:29:06

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

Without an #xy# 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.

#5x^2-3y^2-20x-36y =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 #xy# term and it can be more difficult to tell them apart.