Ellipse — Equation, Foci, Eccentricity & Graph

An ellipse is a graph shaped like a stretched circle. In coordinate geometry, you usually identify it from a standard equation, then read off the center, the long and short half-axes, the foci, and the eccentricity.

Geometrically, an ellipse is the set of points whose distances to two fixed points add up to a constant. Those fixed points are the foci. That definition explains why the graph has a center, a longer direction, and a shorter direction.

For a non-circular ellipse centered at the origin with a horizontal major axis, the standard equation is

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad a > b > 0$$

Here $a$ is the semi-major axis and $b$ is the semi-minor axis. The vertices are $(\pm a, 0)$, and the foci are $(\pm c, 0)$, where

$$c^2 = a^2 - b^2$$

The eccentricity is

$$e = \frac{c}{a}$$

For a non-circular ellipse, $0 < e < 1$. Smaller values of $e$ mean the ellipse is closer to a circle. Values closer to $1$ mean it is more stretched.

Ellipse equation in standard form

The standard forms below are the fastest to read because the ellipse is axis-aligned, not rotated.

If the major axis is horizontal,

$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, \qquad a > b > 0$$

If the major axis is vertical,

$$\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1, \qquad a > b > 0$$

In both cases, $(h, k)$ is the center. For these axis-aligned standard forms, the larger denominator tells you the major-axis direction.

You can read the key parts this way:

• Center: $(h, k)$
• Semi-major axis: $a$
• Semi-minor axis: $b$
• Major-axis direction: the variable under the larger denominator

The foci lie on the major axis, not at the vertices. Their distance from the center is $c$, where

$$c^2 = a^2 - b^2$$

So the focus coordinates are:

• Horizontal major axis: $(h \pm c, k)$
• Vertical major axis: $(h, k \pm c)$

What the foci and eccentricity tell you

The numbers $a$ and $b$ tell you how far the ellipse extends in its long and short directions. The value $c$ tells you how far the foci sit from the center.

If the foci are close to the center, the ellipse looks rounder. If they are farther apart, the ellipse looks narrower. Eccentricity, $e = c/a$, turns that idea into one number.

Worked example: graph $x^2/25 + y^2/9 = 1$

Start with the equation

$$\frac{x^2}{25} + \frac{y^2}{9} = 1$$

Because $25 > 9$, the major axis is horizontal. Now read off

$$a^2 = 25 \Rightarrow a = 5, \qquad b^2 = 9 \Rightarrow b = 3$$

Now find $c$:

$$c^2 = a^2 - b^2 = 25 - 9 = 16$$

so

$$c = 4$$

So the important points are:

• Center: $(0, 0)$
• Vertices: $(\pm 5, 0)$
• Co-vertices: $(0, \pm 3)$
• Foci: $(\pm 4, 0)$

The eccentricity is

$$e = \frac{c}{a} = \frac{4}{5}$$

To sketch the graph, plot the center first, then the vertices and co-vertices. Draw a smooth curve through those four endpoints. Since the major axis is horizontal, the ellipse should be wider than it is tall.

How to graph an ellipse step by step

Put the equation into standard form first. That condition matters because shortcuts like "the larger denominator gives the major axis" only work cleanly for axis-aligned standard form.

Then:

1. Find the center $(h, k)$.
2. Identify $a$ and $b$, with $a > b > 0$ for a non-circular ellipse.
3. Use the larger denominator to identify the major-axis direction.
4. Mark the vertices and co-vertices from the center.
5. If needed, compute $c$ from $c^2 = a^2 - b^2$ and place the foci on the major axis.

If the ellipse is centered at $(h, k)$ instead of the origin, the same steps work after shifting every key point by $(h, k)$.

Common mistakes

Mixing up $a$ and $b$

For a non-circular ellipse in standard form, $a$ is the semi-major axis, so $a > b$. Students sometimes assign $a$ to the $x$ term automatically, but that is only true when the major axis is horizontal.

Using the wrong relation for the foci

For an ellipse, $c^2 = a^2 - b^2$, not $a^2 + b^2$. The wrong sign gives the wrong foci and the wrong eccentricity.

Confusing vertices with foci

The vertices are the endpoints of the major axis. The foci are inside the ellipse unless the shape becomes the circular limit. They are not the same points.

Overusing the denominator shortcut

The larger denominator identifies the major axis only after the equation is in the standard axis-aligned form. A rotated ellipse does not read that way directly.

When ellipses are used

Ellipses appear throughout analytic geometry and conic sections because they connect a geometric definition to an equation you can graph. They also show up in physics models. For example, in the idealized two-body model, orbital paths are ellipses with one focus at the central body.

In class, you most often use ellipses to graph conics, find foci and eccentricity, and compare how the shape changes as $a$, $b$, and $e$ change.

Try a shifted ellipse next

Take

$$\frac{(x-2)^2}{16} + \frac{(y+1)^2}{4} = 1$$

and find the center, vertices, foci, and eccentricity before you sketch it. If you want one more check, compare your graph with the example above and see exactly how the shift changes the key points without changing the overall shape.