Conic Sections

Standard forms of the parabola, ellipse, and hyperbola.

The idea

A conic section is a curve in which a plane cuts a double cone. A level cut gives a circle; tilting the plane gives an ellipse; tilting it parallel to the cone's side gives a parabola; tilting further, so that the plane meets both halves of the cone, gives a hyperbola.

Each curve also has a description by distances, using special points called foci. An ellipse is the set of points whose distances to two foci add to a fixed total; a hyperbola is the set whose distances to two foci differ by a fixed amount; a parabola is the set of points equally far from one focus and a fixed line, its directrix.

Center the curve at the origin with its axis along a coordinate axis and its equation takes a standard form: $x^{2} = 4py, \qquad \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, \qquad \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.$ The squared terms name the curve: one squared variable is a parabola, two added an ellipse, two subtracted a hyperbola. The constants place the rest: $p$ is the distance from the parabola's vertex to its focus, $a$ and $b$ are the distances from the center to the curve along the two axes, and the foci lie at distance $c$ from the center, where $c^{2} = a^{2} - b^{2}$ for the ellipse and $c^{2} = a^{2} + b^{2}$ for the hyperbola.

Ways to work on it

• Walkthrough. Identify a conic and read its axes and foci from standard form.
• Practice. Identify the conic type from its equation.
• Hardest. Find a parabola's directrix and a hyperbola's foci.

Not sure where to start? Take the ten-question placement test.