Ellipses
A planet tracing its orbit, a whispering gallery where a sound made at one point carries clearly to a partner standing far across the room, the oval shadow a tilted glass of water casts: each is an ellipse. An ellipse is the set of all points in a plane whose distances to two fixed points, the foci, add up to the same constant. That single rule turns into a clean equation you can read at a glance.
By the end of this page you can:
- Read an ellipse equation in standard form and name a, b, and c.
- Find the vertices, the endpoints of the minor axis, and the foci.
- Sketch the ellipse from those points.
The floor. For \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with \(a > b\), the larger denominator sits under the variable along the major axis, the vertices are \((\pm a, 0)\), and the foci are at \((\pm c, 0)\) where \(c^2 = a^2 - b^2\).
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart 7e |
| Chapter | 11 |
| Section | 11.2 Ellipses |
| Open alternate | OpenStax Precalculus 2e, 12.1 The Ellipse |
| Difficulty | Core |
| Time | 30 min |
Before You Start
The Idea
Pin two foci on the plane and pull a loop of string taut around them with a pencil. As the pencil moves it keeps the total distance to the two foci constant, and it traces an ellipse. Set the foci on the x-axis at \((\pm c, 0)\), call the constant sum \(2a\), and the algebra collapses to a tidy standard form.
The number a is the distance from the center to a vertex along the major (longer) axis. The number b is the distance from the center to an endpoint of the minor (shorter) axis. The foci live on the major axis, closer to the center than the vertices, at distance c. The three are tied together by \(c^2 = a^2 - b^2\), so the foci are always inside the ellipse.
Key Formulas
An ellipse centered at the origin with foci on the x-axis:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad a > b > 0 \]
with vertices at \((\pm a, 0)\), minor-axis endpoints at \((0, \pm b)\), and foci at \((\pm c, 0)\) where
\[ c^2 = a^2 - b^2. \]
If instead the larger denominator sits under \(y^2\), the major axis is vertical: vertices at \((0, \pm a)\) and foci at \((0, \pm c)\).
Source: Stewart 7e, Section 11.2 Ellipses.
Worked Example
Find the vertices, the minor-axis endpoints, and the foci of
\[ \frac{x^2}{25} + \frac{y^2}{9} = 1, \]
then describe its shape.
Try it before opening the answer. Identify which denominator is larger, take square roots, and use \(c^2 = a^2 - b^2\).
Show the answer
The larger denominator, 25, sits under \(x^2\), so the major axis is horizontal.
- \(a^2 = 25\), so \(a = 5\). Vertices: \((\pm 5, 0)\).
- \(b^2 = 9\), so \(b = 3\). Minor-axis endpoints: \((0, \pm 3)\).
- \(c^2 = a^2 - b^2 = 25 - 9 = 16\), so \(c = 4\). Foci: \((\pm 4, 0)\).
The ellipse is centered at the origin, stretches 5 units left and right, 3 units up and down, and its two foci sit on the x-axis 4 units from center.
Connections
Ellipses build on the distance formula, since the defining rule is a statement about the sum of two distances, and on the parabola, the previous member of the conic family defined by foci. The same standard-form reading carries forward to shifted conics, where the center moves off the origin to \((h, k)\) and x and y are replaced by \(x - h\) and \(y - k\). The hyperbola, the remaining conic, uses the same a, b, c machinery but with a difference of distances rather than a sum, so its sign in the c relation flips.
Resources
- Stewart 7e, Section 11.2 Ellipses (primary).
- OpenStax Precalculus 2e, Section 12.1 The Ellipse (open alternate).
Last updated: 2026-06-24