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):

Vertical ellipse (taller than wide):

Key Components

Center: — the midpoint of the ellipse
Vertices: The endpoints of the major axis, at distance from center
Co-vertices: The endpoints of the minor axis, at distance from center
Foci: Two special points inside the ellipse, at distance from center

The Fundamental Relationship

where:

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

Try it now

What shape is an ellipse most similar to?

Worked Examples

For the ellipse , find the center, vertices, co-vertices, and foci.

Identify the center

The equation is in standard form with and → Center:

Identify and

(larger denominator), → ,

Determine orientation

Since is under , the major axis is horizontal → Horizontal ellipse

Find vertices

Vertices are at → Vertices: and

Find co-vertices

Co-vertices are at → Co-vertices: and

Calculate

, so →

Find foci

Foci are at → Foci: and

Answer: Center: , Vertices: , Co-vertices: , Foci:

Common Mistakes

Using (Pythagorean theorem) instead of

Why it's wrong: The relationship is for hyperbolas, not ellipses. For ellipses, the foci are inside the curve, so .

Correct: For ellipses: . For hyperbolas: .

Confusing which denominator is and which is

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

Correct: Always identify the larger denominator first — that's . Its position (under or term) tells you the orientation.

Forgetting to take the square root when finding , , or

Why it's wrong: The equation gives and , not and directly.

Correct: If , then (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 and .

Interactive Visual

Linear Function Explorer

y = x

Slope (m)1

Y-Intercept (b)0

run

rise

Numerator:

Denominator:

3/4= 75%

Click on the bar to change the fraction

Interactive Sandbox

Interactive Grapher

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

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History

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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 million km and eccentricity , making it nearly circular but technically elliptical.

1Try It Yourself

A comet has an elliptical orbit with semi-major axis AU and semi-minor axis AU.

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

Step 1: Write the mathematical expression

Use :

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 using :

Key Takeaways

1An ellipse is a stretched circle with equation (horizontal) or (vertical)
2The center is at , with always being the larger value (semi-major axis)
3The relationship between , , and (focus distance) is:
4Vertices are at distance from center along the major axis; co-vertices at distance along the minor axis
5Foci are at distance 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 ()
Major axis: The longest diameter of an ellipse, passing through both foci; has length
Minor axis: The shortest diameter of an ellipse, perpendicular to the major axis; has length
Semi-major axis: Half the major axis; the distance from center to a vertex; denoted
Semi-minor axis: Half the minor axis; the distance from center to a co-vertex; denoted
Vertices: The two endpoints of the major axis, at distance from the center
Co-vertices: The two endpoints of the minor axis, at distance from the center
Eccentricity: The ratio measuring how elongated an ellipse is; for ellipses,

Formula Card

Standard Form (horizontal)

where $a > b$

Standard Form (vertical)