Introduction to Ellipses

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

Advanced25 minLesson
Printable worksheet

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):
(x−h)2a2+(y−k)2b2=1where a>b\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \quad \text{where } a > b
Vertical ellipse (taller than wide):
(x−h)2b2+(y−k)2a2=1where a>b\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \quad \text{where } a > b

Key Components

  • Center: (h,k)(h, k), the midpoint of the ellipse
  • Vertices: The endpoints of the major axis, at distance aa from center
  • Co-vertices: The endpoints of the minor axis, at distance bb from center
  • Foci: Two special points inside the ellipse, at distance cc from center

The Fundamental Relationship

c2=a2−b2c^2 = a^2 - b^2
where:
  • aa = semi-major axis (larger value)
  • bb = semi-minor axis (smaller value)
  • cc = distance from center to each focus

Try it now

What shape is an ellipse most similar to?

Worked Examples

For the ellipse x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1, find the center, vertices, co-vertices, and foci.
1

Identify the center

The equation is in standard form with h=0h = 0 and k=0k = 0 → Center: (0,0)(0, 0)
2

Identify a2a^2 and b2b^2

a2=25a^2 = 25 (larger denominator), b2=9b^2 = 9 → a=5a = 5, b=3b = 3
3

Determine orientation

Since a2=25a^2 = 25 is under x2x^2, the major axis is horizontal → Horizontal ellipse
4

Find vertices

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

Find co-vertices

Co-vertices are at (0,±b)=(0,±3)(0, \pm b) = (0, \pm 3) → Co-vertices: (0,−3)(0, -3) and (0,3)(0, 3)
6

Calculate cc

c2=a2−b2=25−9=16c^2 = a^2 - b^2 = 25 - 9 = 16, so c=4c = 4
7

Find foci

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

Common Mistakes

Using c2=a2+b2c^2 = a^2 + b^2 (Pythagorean theorem) instead of c2=a2−b2c^2 = a^2 - b^2

Why it's wrong: The relationship c2=a2+b2c^2 = a^2 + b^2 is for hyperbolas, not ellipses. For ellipses, the foci are inside the curve, so c<ac < a.

Correct: For ellipses: c2=a2−b2c^2 = a^2 - b^2. For hyperbolas: c2=a2+b2c^2 = a^2 + b^2.

Confusing which denominator is a2a^2 and which is b2b^2

Why it's wrong: By convention, aa is always the larger value. The position of a2a^2 determines orientation, not which variable it's under.

Correct: Always identify the larger denominator first, that's a2a^2. Its position (under xx or yy term) tells you the orientation.

Forgetting to take the square root when finding aa, bb, or cc

Why it's wrong: The equation gives a2a^2 and b2b^2, not aa and bb directly.

Correct: If a2=25a^2 = 25, then a=5a = 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±c,k)(h \pm c, k) and (h±a,k)(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

Interactive Visual

Function graph

-10-50510-505xyCF₁F₂
  • c = √|a² − b²| = 4
(x − h)²/a² + (y − k)²/b² = 1
  • Center: (0, 0)
  • Foci: (-4, 0), (4, 0)
  • Vertices: (-5, 0), (5, 0)

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = 3*sqrt(1 - x^2/25)

y = -3*sqrt(1 - x^2/25)

Expression Calculator

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Practice Problems

17 problems
Problem 1 of 17
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≈149.6a \approx 149.6 million km and eccentricity e≈0.017e \approx 0.017, making it nearly circular but technically elliptical.

1Try It Yourself

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

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

Step 1: Write the mathematical expression

Use c2=a2−b2c^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 bb using c2=a2−b2c^2 = a^2 - b^2:

Key Takeaways

  • 1An ellipse is a stretched circle with equation (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 (horizontal) or (x−h)2b2+(y−k)2a2=1\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 (vertical)
  • 2The center is at (h,k)(h, k), with aa always being the larger value (semi-major axis)
  • 3The relationship between aa, bb, and cc (focus distance) is: c2=a2−b2c^2 = a^2 - b^2
  • 4Vertices are at distance aa from center along the major axis; co-vertices at distance bb along the minor axis
  • 5Foci are at distance cc 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.
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.
When a=ba = b, the ellipse becomes a circle. In this case, c=0c = 0 (since c2=a2−b2=0c^2 = a^2 - b^2 = 0), meaning the two foci merge into a single point, the center.
Eccentricity e=cae = \frac{c}{a} measures how "stretched" an ellipse is. For ellipses, 0≤e<10 \leq e < 1. A circle has e=0e = 0 (not stretched), while values close to 1 indicate a very elongated ellipse.

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 (2a2a)
Major axis
The longest diameter of an ellipse, passing through both foci; has length 2a2a
Minor axis
The shortest diameter of an ellipse, perpendicular to the major axis; has length 2b2b
Semi-major axis
Half the major axis; the distance from center to a vertex; denoted aa
Semi-minor axis
Half the minor axis; the distance from center to a co-vertex; denoted bb
Vertices
The two endpoints of the major axis, at distance aa from the center
Co-vertices
The two endpoints of the minor axis, at distance bb from the center
Eccentricity
The ratio e=c/ae = c/a measuring how elongated an ellipse is; for ellipses, 0≤e<10 \leq e < 1

Formula Card

Standard Form (horizontal)

(x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

where $a > b$

Standard Form (vertical)

(x−h)2b2+(y−k)2a2=1\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1

where $a > b$

Focus relationship

c2=a2−b2c^2 = a^2 - b^2

$c$ = distance from center to focus

Eccentricity

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

$0 \leq e < 1$ for ellipses

Today's challengeOne problem, three tries

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