Lesson 15.3

The Ellipse: Major and Minor Axes

Analyzing the foci and the standard equation of an ellipse.

Introduction

An ellipse is the set of all points where the sum of distances to two fixed points (the foci) is constant. It looks like a stretched circle.

1

Prerequisite Connection

You understand focus-directrix (parabola) and can identify conic types.

2

Today's Increment

We learn the standard form, identify major/minor axes, locate foci, and calculate eccentricity.

3

Why This Matters

Planetary orbits are ellipses. Understanding ellipses helps us predict where planets will be!

Major and Minor Axes

Standard Form (Center at Origin)

where (horizontal major axis)

Key Relationship

= distance from center to each focus

Eccentricity

(closer to 0 = more circular)

Terminology

  • • Major axis: Longer axis (length = )
  • • Minor axis: Shorter axis (length = )
  • • Vertices: Endpoints of major axis
  • • Co-vertices: Endpoints of minor axis
  • • Foci: Two points on major axis, units from center

Interactive: Adjust a and b

a = 4b = 2c ≈ 3.46e ≈ 0.87

Worked Examples

Example 1: Finding Foci

Find the foci of .

Step 1: Identify a² and b²

, . Since 25 > 9, major axis is horizontal.

Step 2: Calculate c

Solution

Foci: and

Example 2: Writing the Equation

Write the equation of an ellipse with center at origin, vertices at , and foci at .

Step 1: Identify a and c

Vertices at means and major axis is vertical.
Foci at means .

Step 2: Find b

Solution

Example 3: Translated Ellipse

Find the center, vertices, and foci of .

Step 1: Identify center

Center:

Step 2: Find a, b, c

, , so , .

Solution

Center:
Vertices: and
Foci:

Common Pitfalls

Confusing a² and b²

is ALWAYS the larger value. It determines where the major axis lies.

Wrong formula for c

For ellipses: (subtract!). For hyperbolas it's addition.

Foci on wrong axis

Foci are always on the major axis—the one with the larger denominator.

Real-World Application

Whispering Galleries

Some domed buildings have elliptical cross-sections. A whisper at one focus can be heard clearly at the other focus—even across a large room!

Famous examples include the U.S. Capitol Building and St. Paul's Cathedral. Sound waves reflect off the elliptical ceiling and converge at the opposite focus.

Practice Quiz

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