Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
A conic is a curve obtained by intersecting a plane with a double-napped cone. The types of conics include ellipses, hyperbolas, and parabolas.
5 Must Know Facts For Your Next Test
A circle is a special case of an ellipse where the eccentricity is zero.
The standard form of an ellipse's equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
The standard form of a hyperbola’s equation is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ or $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.
In polar coordinates, the general equation for conics can be expressed as $r = \frac{ed}{1 + e\cos(\theta)}$ where $e$ is the eccentricity and $d$ is the directrix.
The eccentricity ($e$) determines the shape of the conic: if $0 < e < 1$, it’s an ellipse; if $e = 1$, it’s a parabola; if $e > 1$, it’s a hyperbola.
A parameter associated with conic sections that describes their shape; denoted as $e$, it helps classify whether the conic is an ellipse, parabola, or hyperbola.