Classification of conics by the discriminant

Builds on Conic section Ellipse Parabola Hyperbola Circle

Statement

Let $Q = \{\, (x, y) \in \mathbb{R}^2 : A x^2 + B x y + C y^2 + D x + E y + F = 0 \,\}$ be a non-degenerate conic section. Let $\Delta = B^2 - 4AC$ be the discriminant. Then $Q$ is

an ellipse if and only if $\Delta < 0$ ,
a parabola if and only if $\Delta = 0$ ,
a hyperbola if and only if $\Delta > 0$ .

Moreover, $Q$ is a circle if and only if $A = C$ and $B = 0$ .

Remarks

The discriminant is $-4$ times the determinant of $\begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix}$ , the matrix of the quadratic part. A rotation or a translation of the coordinates does not change $\Delta$ or $A + C$ . So the type of a conic does not depend on the choice of Cartesian coordinates. A hyperbola with $A + C = 0$ is rectangular: its asymptotes are perpendicular.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Circle concept
Conic section concept
Ellipse concept
Hyperbola concept
Parabola concept
Classification of conics by the discriminant statement

Tag	`0N5NR32NQG`
URI	`m2://0N5NR32NQG`
Kind	statement, theorem
Writing	W0 stub
MSC	51N20
Status	This copy of the site has no build data. Run `m2 status 0N5NR32NQG`.
Uses	Conic section, Ellipse, Parabola, Hyperbola, Circle

Classification of conics by the discriminant

Statement

Remarks

Reading path

Further reading

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