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Teacher Guide: Introduction to Ellipses

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

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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 c2=a2−b2c^2 = a^2 - b^2 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 aa and bb get closer in value?
Common Misconceptions

Thinking the foci are at the ends of the major axis

Remediation: Show that vertices are at the ends of the major axis, and foci are INSIDE the ellipse at distance cc from center. Calculate cc to show it's always less than aa.

Believing aa must always be under the xx-term

Remediation: Emphasize that aa is defined as the larger value, period. Its position determines orientation: under xx = horizontal, under yy = vertical.

Confusing ellipse and hyperbola focus formulas

Remediation: Create a comparison chart: Ellipse uses subtraction (c2=a2−b2c^2 = a^2 - b^2) because foci are inside. Hyperbola uses addition because foci are outside.

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: x=acos⁡(t)x = a\cos(t), y=bsin⁡(t)y = b\sin(t)
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 aa, bb, 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):
(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

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).

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

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 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.

What is eccentricity?

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

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