Ellipses and Their Equations

An ellipse is a type of conic section. Geometrically, it is the set of all points in a plane where the sum of the distances to two fixed points (called the foci) is a constant.

The Standard Equation of an Ellipse

The standard form of an ellipse centered at the origin $(0,0)$ depends on whether its longest axis (the major axis) is horizontal or vertical.

Horizontal Major Axis: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (Here, $a > b > 0$. The major axis lies on the x-axis.)

Vertical Major Axis: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ (Here, $a > b > 0$. The major axis lies on the y-axis.)

If the center is translated to a point $(h, k)$, you simply replace $x$ with $(x-h)$ and $y$ with $(y-k)$.

Key Features of an Ellipse

To graph or analyze an ellipse, you need to identify four main features:

• Center: The midpoint of the ellipse, $(0,0)$ or $(h,k)$.
• Vertices: The endpoints of the major (longer) axis. The distance from the center to each vertex is $a$.
• Co-vertices: The endpoints of the minor (shorter) axis. The distance from the center to each co-vertex is $b$.
• Foci: Two fixed points located inside the ellipse along the major axis. The distance from the center to each focus is $c$.

You can find the focal distance $c$ using the fundamental relationship for ellipses: $c^2 = a^2 - b^2$

Example Problems

Example 1: Find the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$

1. Identify $a^2$ and $b^2$. Since $25 > 9$, $a^2 = 25$ and $b^2 = 9$.
2. The larger denominator is under $x^2$, so the major axis is horizontal. The center is $(0,0)$.
3. Find $c^2$ using the formula: $c^2 = a^2 - b^2 = 25 - 9 = 16$.
4. Solve for $c$: $c = 4$.
5. Because the major axis is horizontal, the foci are located $c$ units left and right of the center: $(\pm 4, 0)$.

Example 2: Write the equation of an ellipse with vertices $(\pm 5, 0)$ and foci $(\pm 3, 0)$

1. The vertices and foci lie on the x-axis, meaning the center is $(0,0)$ and the major axis is horizontal.
2. The distance to the vertices gives us $a$: $a = 5$, so $a^2 = 25$.
3. The distance to the foci gives us $c$: $c = 3$, so $c^2 = 9$.
4. Find $b^2$ using $c^2 = a^2 - b^2$: $9 = 25 - b^2 \implies b^2 = 16$
5. Write the standard equation by plugging in $a^2$ and $b^2$: $\frac{x^2}{25} + \frac{y^2}{16} = 1$