11.3 Ellipses

Ellipse Fundamentals

An ellipse is the set of all points where the sum of the distances to two fixed points (called foci) is constant. Think of it as a stretched circle. Understanding how to write ellipse equations, identify their parts, and graph them is the core of this section.

Standard Form Equations

Centered at the origin $(0, 0)$:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Here, $a$ and $b$ are the lengths of the semi-axes (half the full axis length). Which axis is "major" depends on which denominator is larger:

• If $a > b$, the major axis is horizontal (stretches along the x-axis)
• If $b > a$, the major axis is vertical (stretches along the y-axis)

The larger denominator always sits under the variable whose axis is the major axis.

Centered at $(h, k)$:

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

This is the same ellipse, just shifted so its center sits at $(h, k)$ instead of the origin. The shape and orientation stay the same for identical $a$ and $b$ values.

Components of Ellipses

• Center: The point $(h, k)$ at the middle of the ellipse.
• Vertices: The endpoints of the major axis. These are the points on the ellipse farthest from the center.
• Co-vertices: The endpoints of the minor axis. These are the points closest to the center (measured along an axis).
• Major axis: The longest diameter, passing through the center and both vertices.
• Minor axis: The shortest diameter, perpendicular to the major axis, passing through the center and both co-vertices.
• Foci (plural of focus): Two points inside the ellipse, located on the major axis, equidistant from the center. The distance from the center to each focus is:

$c = \sqrt{a^2 - b^2}$

where $a$ is the semi-major axis length and $b$ is the semi-minor axis length. Always make sure $a > b$ when using this formula.

• Eccentricity: Measures how "stretched" the ellipse is, calculated as $e = \frac{c}{a}$. This value falls between 0 and 1. When $e = 0$, the ellipse is a perfect circle. As $e$ approaches 1, the ellipse gets more elongated. For reference, Earth's orbital eccentricity is about 0.0167, which is very close to circular.

Graphing Ellipses

To graph an ellipse from its equation:

1. Identify the center $(h, k)$ from the equation. If there's no $(x - h)$ or $(y - k)$ shift, the center is $(0, 0)$.

2. Find $a$ and $b$ by taking the square roots of the denominators. The larger value is $a$ (semi-major), the smaller is $b$ (semi-minor).

3. Determine orientation. If the larger denominator is under $x^2$, the major axis is horizontal. If it's under $y^2$, the major axis is vertical.

4. Plot the vertices by moving $a$ units from the center along the major axis in both directions.

5. Plot the co-vertices by moving $b$ units from the center along the minor axis in both directions.

6. Sketch the curve through all four points, forming a smooth oval.

To also plot the foci, calculate $c = \sqrt{a^2 - b^2}$ and mark points $c$ units from the center along the major axis.

Writing Equations of Ellipses

Given the center, $a$, and $b$: Plug directly into the standard form.

Given vertices and co-vertices:

1. Find the center by calculating the midpoint of the two vertices (or the two co-vertices).
2. Find $a$ by measuring the distance from the center to either vertex.
3. Find $b$ by measuring the distance from the center to either co-vertex.
4. Substitute $(h, k)$, $a^2$, and $b^2$ into the standard form equation.

Given the foci and a vertex (or the sum of distances):

1. Find the center as the midpoint of the two foci.

2. Determine $c$ (distance from center to a focus) and $a$ (distance from center to a vertex).

3. Solve for $b$ using $b = \sqrt{a^2 - c^2}$.

4. Write the equation in standard form.

Ellipse Applications

Real-World Examples

Planetary orbits: Planets travel in elliptical paths with the Sun at one focus. This is Kepler's First Law of Planetary Motion. The other focus is an empty point in space.

Whispering galleries: Rooms with elliptical ceilings or walls (like the U.S. Capitol Building) have a striking acoustic property. A whisper spoken at one focus reflects off the curved surface and converges at the other focus, making it clearly audible across the room.

Medical and engineering uses: The reflective property of ellipses (waves from one focus reflect to the other) is used in lithotripsy, a medical procedure that breaks up kidney stones with focused sound waves. Satellite dishes and some antenna designs also rely on this principle.

Conic Sections Context

Ellipses belong to the family of conic sections, curves formed by slicing a double cone with a plane. The other conics are circles (a special case of the ellipse where $a = b$), parabolas, and hyperbolas.

A few related terms you may encounter:

• Directrix: A reference line used in an alternate definition of conics. For any point on the ellipse, the ratio of its distance to a focus over its distance to the directrix equals the eccentricity $e$.
• Latus rectum: A chord that passes through a focus and runs perpendicular to the major axis. Its length is $\frac{2b^2}{a}$.