Conic Sections Worked Examples Cheat Sheet

Key Facts

• A circle with center $(h,k)$ and radius $r$ has equation $(x-h)^2+(y-k)^2=r^2$.
• A vertical parabola with vertex $(h,k)$ has equation $(x-h)^2=4p(y-k)$, focus $(h,k+p)$, and directrix $y=k-p$.
• A horizontal parabola with vertex $(h,k)$ has equation $(y-k)^2=4p(x-h)$, focus $(h+p,k)$, and directrix $x=h-p$.
• An ellipse centered at $(h,k)$ with horizontal major axis has equation $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$, where $a>b$.
• For an ellipse, the foci satisfy $c^2=a^2-b^2$, and the foci lie on the major axis.
• A horizontal hyperbola centered at $(h,k)$ has equation $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$ and asymptotes $y-k=\pm \frac{b}{a}(x-h)$.
• A vertical hyperbola centered at $(h,k)$ has equation $\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$ and asymptotes $y-k=\pm \frac{a}{b}(x-h)$.
• In a general quadratic equation $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ with $B=0$, same-sign $A$ and $C$ suggest an ellipse or circle, opposite signs suggest a hyperbola, and exactly one squared term suggests a parabola.

Vocabulary

Conic section

A conic section is a curve formed by the intersection of a plane and a double cone.

Focus

A focus is a fixed point used to define or locate a conic, such as the point inside a parabola, ellipse, or hyperbola.

Directrix

A directrix is a fixed line used with a focus to define a parabola as the set of points equally distant from both.

Vertex

A vertex is a key turning point of a conic, such as the endpoint of a parabola or one of the closest points on a hyperbola.

Major axis

The major axis is the longer central axis of an ellipse, passing through its center, vertices, and foci.

Asymptote

An asymptote is a line that a hyperbola approaches but does not touch as its branches extend.