Conic Sections 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 has equation $(x-h)^2=4p(y-k)$, vertex $(h,k)$, focus $(h,k+p)$, and directrix $y=k-p$.
• A horizontal parabola has equation $(y-k)^2=4p(x-h)$, vertex $(h,k)$, 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 $a>b$ and $c^2=a^2-b^2$.
• A hyperbola centered at $(h,k)$ opening left and right has equation $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$, vertices $(h\pm a,k)$, and $c^2=a^2+b^2$.
• A hyperbola centered at $(h,k)$ opening up and down has equation $\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$, vertices $(h,k\pm a)$, and $c^2=a^2+b^2$.
• For an ellipse or hyperbola, eccentricity is $e=\frac{c}{a}$, with $0<e<1$ for ellipses and $e>1$ for hyperbolas.
• The asymptotes of $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$ are $y-k=\pm \frac{b}{a}(x-h)$.

Vocabulary

Conic section

A conic section is a curve formed by intersecting a plane with a double cone.

Focus

A focus is a fixed point used to define a conic, often paired with distance rules involving the curve.

Directrix

A directrix is a fixed line used with a focus to define a parabola by equal distances.

Vertex

A vertex is a turning point or endpoint of a main axis of a conic.

Major axis

The major axis of an ellipse is its longest central axis, with length $2a$.

Asymptote

An asymptote is a line that a hyperbola approaches but never reaches as the graph extends.