Teacher Guide: Ellipses in Standard Form

Learn to write, graph, and analyze ellipses using the standard form equation.

Present LessonView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

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

Write the equation of an ellipse in standard form given its key features
Identify center, vertices, co-vertices, and foci from a standard form equation
Determine whether an ellipse is horizontal or vertical from its equation
Convert general form ellipse equations to standard form by completing the square
Apply the relationship $c^2 = a^2 - b^2$ to find foci

Prerequisites

• Completing the square
• Distance formula
• Graphing on the coordinate plane
• Introduction to ellipses (basic definition)

Discussion Starters

1. Why do you think planetary orbits are ellipses rather than circles?
2. How does changing the value of $a$ vs $b$ affect the shape of the ellipse?
3. What real-world situations might require you to find the foci of an ellipse?
4. If an ellipse has eccentricity close to 0, what does it look like? What about close to 1?

Common Misconceptions

The larger number in the equation always goes with x

Remediation: Show examples of both horizontal and vertical ellipses. Emphasize that $a$ is simply the larger of the two values, regardless of position.

Foci are on the minor axis

Remediation: Use the string definition of an ellipse. Have students draw strings from foci to points on the ellipse to see the constant sum property.

Differentiation Ideas

For Struggling Students:

• Start with ellipses centered at the origin
• Use color-coding to identify $a$ and $b$ in equations
• Provide formula reference cards during practice

For On-Level Students:

• Practice both horizontal and vertical ellipses
• Convert general form to standard form
• Find all key features from equations

For Advanced Students:

• Explore eccentricity and its meaning
• Derive the standard form from the focus definition
• Investigate reflective properties 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
HSG-GPE.A.1 (CCSS.MATH.CONTENT.HSG.GPE.A.1)
Derive the equation of a circle; complete the square to find the center and radius

Lesson Resources

visualInteractive Ellipse Explorer
Adjust center, a, and b to see how the ellipse changes
activityEllipse Component Matching
Match equations to their graphs and key features
worksheetConverting to Standard Form
Practice completing the square with ellipse equations

Lesson Content

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

Definition

An ellipse is the set of all points

(x, y)

such that the sum of the distances from two fixed points (called foci) is constant.

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: $(h, k)$
Semi-major axis: $a$ (always the larger value)
Semi-minor axis: $b$ (always the smaller value)
Relationship: $c^2 = a^2 - b^2$ where $c$ is the distance from center to each focus

Worked Examples

Find the center, vertices, co-vertices, and foci of the ellipse: $\frac{(x-2)^2}{25} + \frac{(y+3)^2}{9} = 1$

Identify the center $(h, k)$

From $(x-2)^2$ and $(y+3)^2$ , we have $h = 2$ and $k = -3$ → Center: $(2, -3)$

Find $a^2$ and $b^2$

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

Determine orientation

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

Find vertices (on major axis)

$(h \pm a, k) = (2 \pm 5, -3)$ → Vertices: $(-3, -3)$ and $(7, -3)$

Find co-vertices (on minor axis)

$(h, k \pm b) = (2, -3 \pm 3)$ → Co-vertices: $(2, -6)$ and $(2, 0)$

Calculate $c$ for foci

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

Find foci (on major axis)

$(h \pm c, k) = (2 \pm 4, -3)$ → Foci: $(-2, -3)$ and $(6, -3)$

Answer: Center: $(2, -3)$ ; Vertices: $(-3, -3)$ , $(7, -3)$ ; Co-vertices: $(2, -6)$ , $(2, 0)$ ; Foci: $(-2, -3)$ , $(6, -3)$

Common Mistakes

Confusing which denominator gives $a^2$ vs $b^2$

Why it's wrong: Students assume $a^2$ is always under $x^2$ , but $a > b$ by definition, so $a^2$ is the larger denominator.

Correct: Always identify: $a^2$ = larger denominator, $b^2$ = smaller denominator. Then determine orientation based on which variable has $a^2$ .

Using $c^2 = a^2 + b^2$ (hyperbola formula) instead of $c^2 = a^2 - b^2$

Why it's wrong: Hyperbolas use addition, ellipses use subtraction. This is a critical difference.

Correct: For ellipses: $c^2 = a^2 - b^2$ . Remember: foci are inside the ellipse, so $c < a$ .

Forgetting to change signs when reading center from equation

Why it's wrong: The equation has $(x - h)^2$ , so $(x + 3)^2$ means $h = -3$ , not $h = 3$ .

Correct: $(x - h)^2$ with $(x + 3)^2$ means $-h = 3$ , so $h = -3$ .

Placing foci on the minor axis

Why it's wrong: Students may forget that foci are always on the major axis (the longer one).

Correct: Foci are always on the major axis, at distance $c$ from the center.

Why It Matters

Ellipses appear throughout science and engineering:

Astronomy: Planets orbit the Sun in elliptical paths (Kepler's First Law)
Architecture: Whispering galleries use elliptical ceilings for acoustic effects
Medicine: Lithotripsy machines use elliptical reflectors to focus sound waves on kidney stones
Optics: Elliptical mirrors reflect all light from one focus to the other

Understanding the standard form helps us analyze and design these systems mathematically.

Real World Applications

Planetary Orbits

All 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.

1Try It Yourself

A comet orbits the Sun in an ellipse with semi-major axis $a = 18$ AU and $c = 17$ AU (distance from center to Sun).

Find the semi-minor axis $b$ .

Step 1: Write the mathematical expression

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

Whispering Galleries

In an elliptical room, sound from one focus reflects to the other focus, allowing whispers to travel across the room.

Example:

The Capitol Building in Washington D.C. has an elliptical room where this effect occurs.

2Try It Yourself

An elliptical hall is 30 meters long and 20 meters wide.

How far apart are the two focal points (whispering spots)?

Step 1: Write the mathematical expression

Find $2c$ where $a = 15$ and $b = 10$ :

Key Takeaways

1An ellipse in standard form is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ (horizontal) or with $a^2$ and $b^2$ swapped (vertical)
2The center is at $(h, k)$ , and $a > b$ always (larger denominator determines orientation)
3Vertices are on the major axis at distance $a$ from center; co-vertices are on minor axis at distance $b$
4Foci are on the major axis at distance $c$ from center, where $c^2 = a^2 - b^2$
5To convert general form to standard form, complete the square for both variables

Frequently Asked Questions

How do I know if an ellipse is horizontal or vertical?

Look at which variable has the larger denominator. If

a^2

(larger) is under

(x-h)^2

, the major axis is horizontal. If

a^2

is under

(y-k)^2

, the major axis is vertical.

What is the difference between a and c?

a

is the distance from center to vertex (on the major axis).

c

is the distance from center to focus. Since foci are inside the ellipse,

c < a

What happens when a = b?

When

a = b

, the ellipse becomes a circle! The equation becomes

(x-h)^2 + (y-k)^2 = a^2

, and there are no distinct foci (

c = 0

Glossary

Ellipse: The set of all points where the sum of distances to two foci is constant
Semi-major axis: Half the length of the longest diameter, denoted $a$
Semi-minor axis: Half the length of the shortest diameter, denoted $b$
Focus (pl. foci): One of two fixed points inside the ellipse; $c^2 = a^2 - b^2$
Vertex: An endpoint of the major axis
Co-vertex: An endpoint of the minor axis
Eccentricity: The ratio $e = c/a$ measuring how elongated the ellipse is (0 = circle, close to 1 = very elongated)

Formula Card

Standard Form (horizontal)

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

$a > b$, major axis horizontal

Standard Form (vertical)

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

$a > b$, major axis vertical

Relationship

c^2 = a^2 - b^2

$c$ = distance from center to focus

Eccentricity

e = \frac{c}{a}

$0 < e < 1$ for all ellipses