Ellipses

An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (the foci) is a constant. If you have ever drawn an oval by pinning two tacks on a board, looping a string around them, and tracing with a pencil, you have drawn an ellipse. Planets orbit the sun in ellipses, whispering galleries use elliptical ceilings to transmit sound, and elliptical arches appear in architecture worldwide.

The Definition and Key Measurements

For any point $P$ on the ellipse with foci $F_1$ and $F_2$:

$d(P, F_1) + d(P, F_2) = 2a$

where $a$ is the semi-major axis (half the length of the longest diameter).

Key measurements:

• $a$ = semi-major axis (vertex to center, along the longer direction)
• $b$ = semi-minor axis (co-vertex to center, along the shorter direction)
• $c$ = distance from center to each focus

The relationship between them is:

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

This means $a > b$ always (the semi-major axis is the longer one), and $c$ is always less than $a$ (the foci are inside the ellipse).

Standard Forms

Horizontal Major Axis (wider than tall)

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

• Center: $(h, k)$
• Vertices: $(h \pm a, k)$ — endpoints of the major axis
• Co-vertices: $(h, k \pm b)$ — endpoints of the minor axis
• Foci: $(h \pm c, k)$ where $c = \sqrt{a^2 - b^2}$

Vertical Major Axis (taller than wide)

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

• Center: $(h, k)$
• Vertices: $(h, k \pm a)$ — endpoints of the major axis
• Co-vertices: $(h \pm b, k)$ — endpoints of the minor axis
• Foci: $(h, k \pm c)$ where $c = \sqrt{a^2 - b^2}$

Key pattern: The larger denominator is always $a^2$. If it is under the $x$-term, the major axis is horizontal. If it is under the $y$-term, the major axis is vertical.

Eccentricity

The eccentricity measures how “elongated” the ellipse is:

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

Since $c$ is always between $0$ and $a$, eccentricity satisfies $0 \le e < 1$:

• $e = 0$: the ellipse is a perfect circle ($c = 0$, foci at center)
• $e$ close to 1: the ellipse is very elongated (foci near the ends)
• Earth’s orbit: $e \approx 0.017$ (nearly circular)
• Pluto’s orbit: $e \approx 0.248$ (noticeably elongated)

Worked Examples

Example 1: Identifying Parts from the Equation

Find the center, vertices, co-vertices, foci, and eccentricity of:

$\frac{(x - 1)^2}{25} + \frac{(y + 3)^2}{9} = 1$

Step 1: Center: $(1, -3)$.

Step 2: Since $25 > 9$, the larger denominator is under $x$, so the major axis is horizontal. $a^2 = 25$ so $a = 5$. $b^2 = 9$ so $b = 3$.

Step 3: $c^2 = a^2 - b^2 = 25 - 9 = 16$, so $c = 4$.

• Vertices: $(1 \pm 5, -3) = (-4, -3)$ and $(6, -3)$
• Co-vertices: $(1, -3 \pm 3) = (1, -6)$ and $(1, 0)$
• Foci: $(1 \pm 4, -3) = (-3, -3)$ and $(5, -3)$
• Eccentricity: $e = \frac{4}{5} = 0.8$

Example 2: Vertical Major Axis

Find the foci of:

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

The larger denominator $16$ is under $y^2$, so the major axis is vertical. $a^2 = 16$, $b^2 = 4$.

$c^2 = 16 - 4 = 12, \quad c = 2\sqrt{3}$

Foci: $(0, \pm 2\sqrt{3}) \approx (0, \pm 3.46)$

Example 3: Writing the Equation from Given Information

Write the equation of an ellipse with center $(0, 0)$, vertices at $(\pm 6, 0)$, and foci at $(\pm 4, 0)$.

Step 1: Vertices at $(\pm 6, 0)$ means $a = 6$ and the major axis is horizontal.

Step 2: Foci at $(\pm 4, 0)$ means $c = 4$.

Step 3: $b^2 = a^2 - c^2 = 36 - 16 = 20$.

$\frac{x^2}{36} + \frac{y^2}{20} = 1$

Example 4: Converting from General Form

Convert $4x^2 + 9y^2 - 16x + 54y + 61 = 0$ to standard form.

Step 1 — Group and factor:

$4(x^2 - 4x) + 9(y^2 + 6y) = -61$

Step 2 — Complete the square:

$4(x^2 - 4x + 4) + 9(y^2 + 6y + 9) = -61 + 16 + 81$

$4(x - 2)^2 + 9(y + 3)^2 = 36$

Step 3 — Divide by 36:

$\frac{(x - 2)^2}{9} + \frac{(y + 3)^2}{4} = 1$

Center: $(2, -3)$, $a^2 = 9$ (under $x$), $b^2 = 4$ (under $y$). Major axis is horizontal. $a = 3$, $b = 2$, $c = \sqrt{5}$.

Graphing an Ellipse

1. Find the center from the equation
2. Identify $a$ and $b$ — the larger denominator is $a^2$
3. Plot the center, then move $a$ units along the major axis in both directions (vertices) and $b$ units along the minor axis (co-vertices)
4. Sketch the oval through these four points
5. Mark the foci at distance $c$ from center along the major axis

Real-World Application: Whispering Gallery

The U.S. Capitol’s National Statuary Hall has an elliptical ceiling. Sound from one focus travels to the other focus with minimal loss — a person whispering at one focus can be heard clearly at the other, even from 40+ feet away.

If the hall’s ceiling is modeled by:

$\frac{x^2}{625} + \frac{y^2}{400} = 1$

then $a = 25$ feet, $b = 20$ feet, and $c = \sqrt{625 - 400} = \sqrt{225} = 15$ feet. The two “sweet spots” for whispering are 15 feet from the center along the major axis.

Common Mistakes

1. Confusing which denominator is $a^2$. Always: $a^2$ is the larger denominator, regardless of whether it is under $x$ or $y$.
2. Using $c^2 = a^2 + b^2$. That is the hyperbola formula. For ellipses: $c^2 = a^2 - b^2$.
3. Forgetting to divide when converting from general form. After completing the square, both sides must be divided so the right side equals 1.
4. Placing foci along the minor axis. Foci are always on the major axis.

Practice Problems

Key Takeaways

• An ellipse is the set of points where the sum of distances to two foci equals $2a$
• Standard form: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ (horizontal) or denominators swapped (vertical) — the larger denominator is always $a^2$
• The relationship $c^2 = a^2 - b^2$ connects semi-major axis, semi-minor axis, and focal distance
• Eccentricity $e = c/a$ measures elongation: $0$ is a circle, close to $1$ is very elongated
• A circle is a special ellipse with $a = b$ (and $c = 0$)

Return to College Algebra for more topics in this section.