Teacher Guide: Introduction to Ellipses

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

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Conic Sections. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Define an ellipse as a conic section and identify its key components
Write and interpret the standard form equation of an ellipse
Calculate the foci, vertices, and co-vertices from an equation
Distinguish between horizontal and vertical ellipses
Apply the relationship to solve problems

Prerequisites

• Understanding of the coordinate plane and graphing
• Familiarity with completing the square
• Knowledge of the Pythagorean theorem
• Basic understanding of conic sections

Discussion Starters

1. Why do you think planets orbit in ellipses rather than perfect circles?
2. If you were designing a stadium, why might an elliptical shape be useful?
3. How does the eccentricity change as an ellipse becomes more circular or more elongated?
4. What happens to the foci as and get closer in value?

Common Misconceptions

Thinking the foci are at the ends of the major axis

Believing must always be under the -term

Confusing ellipse and hyperbola focus formulas

Differentiation Ideas

For Struggling Students:

• Start with ellipses centered at the origin before introducing translations
• Use color-coding: major axis in red, minor in blue, foci in green
• Provide a checklist for finding all components step by step

For On-Level Students:

• Work with translated ellipses (center not at origin)
• Practice converting from general form to standard form
• Solve word problems involving orbital mechanics

For Advanced Students:

• Explore the reflective property of ellipses (why whispers travel between foci)
• Derive the equation of an ellipse from the definition using distance formula
• Investigate parametric equations of ellipses: ,

Standards Alignment

HSG-GPE.A.3 (CCSS.MATH.CONTENT.HSG.GPE.A.3)
Derive the equations of ellipses given the foci, using the fact that the sum of distances from the foci is constant
F-IF.C.7 (CCSS.MATH.CONTENT.HSF.IF.C.7)
Graph functions expressed symbolically and show key features of the graph

Lesson Resources

visualInteractive Ellipse Explorer
Adjust , , and center to see how the ellipse changes
activityString Construction
Use two pins and string to draw an ellipse, demonstrating the focal property
worksheetEllipse Identification
Practice identifying components from equations and graphs

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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

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 .

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

What's the difference between an ellipse and an oval?

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 is an ellipse actually a circle?

When , the ellipse becomes a circle. In this case, (since ), meaning the two foci merge into a single point — the center.

What is eccentricity?

Eccentricity measures how "stretched" an ellipse is. For ellipses, . A circle has (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 ()
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)