Introduction to Ellipses

Learn what an ellipse is, its key features, and the standard form equation.

Advanced25 minLesson

Definition

An ellipse is a conic section formed by a plane intersecting a cone at an angle. It looks like a stretched or compressed circle.

Standard Form Equations

Horizontal ellipse (wider than tall):

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

Vertical ellipse (taller than wide):

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

Key Components

Center: $(h, k)$ — the midpoint of the ellipse
Vertices: The endpoints of the major axis, at distance $a$ from center
Co-vertices: The endpoints of the minor axis, at distance $b$ from center
Foci: Two special points inside the ellipse, at distance $c$ from center

The Fundamental Relationship

c^2 = a^2 - b^2

where:

$a$ = semi-major axis (larger value)
$b$ = semi-minor axis (smaller value)
$c$ = distance from center to each focus

Try it now

What shape is an ellipse most similar to?

Worked Examples

For the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ , find the center, vertices, co-vertices, and foci.

Identify the center

The equation is in standard form with $h = 0$ and $k = 0$ → Center: $(0, 0)$

Identify $a^2$ and $b^2$

$a^2 = 25$ (larger denominator), $b^2 = 9$ → $a = 5$ , $b = 3$

Determine orientation

Since $a^2 = 25$ is under $x^2$ , the major axis is horizontal → Horizontal ellipse

Find vertices

Vertices are at $(\pm a, 0) = (\pm 5, 0)$ → Vertices: $(-5, 0)$ and $(5, 0)$

Find co-vertices

Co-vertices are at $(0, \pm b) = (0, \pm 3)$ → Co-vertices: $(0, -3)$ and $(0, 3)$

Calculate $c$

$c^2 = a^2 - b^2 = 25 - 9 = 16$ , so $c = 4$

Find foci

Foci are at $(\pm c, 0) = (\pm 4, 0)$ → Foci: $(-4, 0)$ and $(4, 0)$

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

Common Mistakes

Using $c^2 = a^2 + b^2$ (Pythagorean theorem) instead of $c^2 = a^2 - b^2$

Why it's wrong: The relationship $c^2 = a^2 + b^2$ is for hyperbolas, not ellipses. For ellipses, the foci are inside the curve, so $c < a$ .

Correct: For ellipses: $c^2 = a^2 - b^2$ . For hyperbolas: $c^2 = a^2 + b^2$ .

Confusing which denominator is $a^2$ and which is $b^2$

Why it's wrong: By convention, $a$ is always the larger value. The position of $a^2$ determines orientation, not which variable it's under.

Correct: Always identify the larger denominator first — that's $a^2$ . Its position (under $x$ or $y$ term) tells you the orientation.

Forgetting to take the square root when finding $a$ , $b$ , or $c$

Why it's wrong: The equation gives $a^2$ and $b^2$ , not $a$ and $b$ directly.

Correct: If $a^2 = 25$ , then $a = 5$ (not 25). Always square root the denominators to get the actual axis lengths.

Mixing up vertices and foci positions for vertical vs horizontal ellipses

Why it's wrong: Students often place foci along the wrong axis.

Correct: Foci and vertices are ALWAYS on the major axis. If the major axis is horizontal, both are at $(h \pm c, k)$ and $(h \pm a, k)$ .

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Ellipses Explained: Vertex, Foci, Eccentricity, and Equations
by Understand The Math
Watch on YouTube

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Practice Problems

17 problems

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Easy

What shape is an ellipse most similar to?

Why It Matters

Ellipses appear throughout nature and technology:

Planetary Orbits: All planets orbit the Sun in elliptical paths (Kepler's First Law)
Architecture: The famous Whispering Gallery in St. Paul's Cathedral uses the reflective property of ellipses
Medical Imaging: Lithotripsy uses elliptical reflectors to break kidney stones without surgery
Astronomy: Satellite orbits, comet paths, and galaxy shapes are all elliptical
Engineering: Elliptical gears, bridges, and stadium designs utilize ellipse properties

Understanding ellipses connects algebra and geometry to real-world phenomena!

Real World Applications

Planetary Orbits

Johannes Kepler discovered that planets orbit the Sun in elliptical paths, with the Sun at one focus.

Example:

Earth's orbit has $a \approx 149.6$ million km and eccentricity $e \approx 0.017$ , making it nearly circular but technically elliptical.

1Try It Yourself

A comet has an elliptical orbit with semi-major axis $a = 20$ AU and semi-minor axis $b = 12$ AU.

How far from the center of the orbit is the Sun (one focus)?

Step 1: Write the mathematical expression

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

Whispering Galleries

In an elliptical room, a whisper at one focus can be heard clearly at the other focus due to reflection properties.

Example:

The National Statuary Hall in the US Capitol has this property — a whisper on one side can be heard 40 feet away at the opposite focus.

2Try It Yourself

An elliptical whispering gallery has a major axis of 80 feet and foci that are 60 feet apart.

What is the length of the minor axis?

Step 1: Write the mathematical expression

Find $b$ using $c^2 = a^2 - b^2$ :

Key Takeaways

1An ellipse is a stretched circle with equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ (horizontal) or $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ (vertical)
2The center is at $(h, k)$ , with $a$ always being the larger value (semi-major axis)
3The relationship between $a$ , $b$ , and $c$ (focus distance) is: $c^2 = a^2 - b^2$
4Vertices are at distance $a$ from center along the major axis; co-vertices at distance $b$ along the minor axis
5Foci are at distance $c$ from center along the major axis, inside the ellipse

Frequently Asked Questions

Mathematically, an ellipse has a precise definition with two foci where the sum of distances from any point to the foci is constant. An oval is a general term for any egg-shaped curve. All ellipses are ovals, but not all ovals are ellipses.

Glossary

Ellipse: A conic section where the sum of distances from any point to two fixed points (foci) is constant
Focus (pl. Foci): One of two special points inside an ellipse; the sum of distances from any point on the ellipse to both foci is constant ( $2a$ )
Major axis: The longest diameter of an ellipse, passing through both foci; has length $2a$
Minor axis: The shortest diameter of an ellipse, perpendicular to the major axis; has length $2b$
Semi-major axis: Half the major axis; the distance from center to a vertex; denoted $a$
Semi-minor axis: Half the minor axis; the distance from center to a co-vertex; denoted $b$
Vertices: The two endpoints of the major axis, at distance $a$ from the center
Co-vertices: The two endpoints of the minor axis, at distance $b$ from the center
Eccentricity: The ratio $e = c/a$ measuring how elongated an ellipse is; for ellipses, $0 \leq e < 1$

Formula Card

Standard Form (horizontal)

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

where $a > b$

Standard Form (vertical)

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