Identifying Conics

Algebra 2 · Axiom Academy

One number — the discriminant B^2 - 4AC — tells you whether a second-degree equation is an ellipse, a parabola, or a hyperbola. 1. One Number Decides the Shape The type of a conic hides in just three coefficients — A , B , and C , the ones riding the squared and cross terms. Fold them into the discriminant and watch its sign : as slides from negative, through zero, to positive, the very same equation bends from an ellipse into a parabola into a hyperbola. The general second-degree equation The discriminant — its sign is the whole story 2. When an Ellipse Is Really a Circle A negative discriminant always means an ellipse — and a circle is the most symmetric ellipse of all. It shows up under two exact conditions at once: the cross term is gone ( B = 0 ) and the squared coefficients match ( A = C ). Drop the tilt, then match the axes, and the ellipse tightens into a perfect circle. No xy term means the conic isn't tilted — its axes line up with the x - and y -axes. Equal coefficients on x^2 and y^2 make the two axes exactly the same length. 3. Classify Any Conic in One Step Here is the whole method: read off A , B , and C , compute , and check its sign. It even handles tilted conics that hide their type. Take xy = 1 — there is no x^2 or y^2 term, so AC = 0 ; yet the cross term makes , and it is a hyperbola. Four quick classifications — read A,B,C , compute the discriminant, check the sign: , so . Equal coefficients, no xy term — a circle . , so . Negative, but — an ellipse .

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