Conic Sections Reference
Ellipses
An ellipse is the set of points whose total distance from two fixed foci is constant.
Solved Examples
Jump directly to a worked example by graph feature or equation form.
Fact Table
| Feature | Horizontal Major Axis | Vertical Major Axis |
|---|---|---|
| Standard form | $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ | $\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$ |
| Center | $(h,k)$ | |
| Foci | $(h\pm c,k)$ | $(h,k\pm c)$ |
| Key relation | $c^2=a^2-b^2$, where $a^2$ is the larger denominator. | |
Content Formulas
Vertices
$$(h\pm a,k)\text{ or }(h,k\pm a)$$
Co-vertices
$$(h,k\pm b)\text{ or }(h\pm b,k)$$
Focal Distance
$$c=\sqrt{a^2-b^2}$$
The larger denominator gives the direction of the major axis. It is not determined by which variable appears first.
Classic Examples
Read an Ellipse from Standard Form
Find the center, vertices, and co-vertices of $\frac{(x-2)^2}{9}+\frac{(y+1)^2}{4}=1$.
Solution Steps
- Read the center from the opposite signs inside the squared terms.
- Use $a=3$ from the larger denominator and move horizontally for the vertices.
- Use $b=2$ and move vertically for the co-vertices.
$$\text{center}=(2,-1)$$$$\text{vertices}=(2\pm3,-1)=(-1,-1),(5,-1)$$$$\text{co-vertices}=(2,-1\pm2)=(2,-3),(2,1)$$
Write an Ellipse from Its Axes
Write the equation of an ellipse centered at the origin with horizontal vertices $(\pm5,0)$ and co-vertices $(0,\pm3)$.
Solution Steps
- Read $a=5$ from the horizontal vertices.
- Read $b=3$ from the vertical co-vertices.
- Use the horizontal-major-axis standard form.
$$a^2=25,\qquad b^2=9$$$$\text{major axis is horizontal}$$$$\boxed{\frac{x^2}{25}+\frac{y^2}{9}=1}$$
Find the Foci
Find the foci of $\frac{x^2}{25}+\frac{y^2}{9}=1$.
Solution Steps
- Identify $a^2=25$ and $b^2=9$ from the denominators.
- Use $c^2=a^2-b^2$.
- Place the foci along the horizontal major axis.
$$a=5,\qquad b=3$$$$c^2=25-9=16\Rightarrow c=4$$$$\boxed{(-4,0)\text{ and }(4,0)}$$