Derive the Equation of an Ellipse

HSG.GPE.A.3 — Lesson 1 of 2

Expressing Geometric Properties with Equations

Learning Objectives

By the end of this lesson you will be able to:

1. Define an ellipse as the set of all points where the sum of distances from two fixed foci equals a constant
2. Derive the standard equation using the distance formula and double-squaring
3. Identify , , , vertices, co-vertices, and foci from an equation in standard form

Hook: A String, Two Pins, and a Pencil

Place two thumbtacks in a board.

Loop a string of fixed length around them.

Pull the string taut with a pencil tip and trace the curve.

Every point the pencil visits satisfies one condition:

This curve is an ellipse.

The Locus Definition

An ellipse is the set of all points in the plane such that:

where and are the two foci and is the constant sum.

Requirement: (the string must be longer than the gap between pins)

The Setup: Foci, Axes, and Point P

Why : A Geometric Proof

Consider the top co-vertex at — the top of the minor axis.

Distances from to each focus:

Apply the locus condition:

M1 Watch-out: This relationship is specific to ellipses. For a hyperbola the sign flips: .

Confirming at the Vertex

Consider the right vertex at — the end of the major axis.

Distance from to :

Distance from to :

Sum: ✓

This confirms is indeed the half-length of the major axis — and that the constant sum equals the major axis length.

Which Direction Is the Major Axis?

M2 Watch-out: is always the semi-major axis, but it is not always under .

Rule: The major axis runs along whichever variable has the larger denominator.

When the larger denominator is under : is that denominator, vertices are at , foci at .

Check-In

An ellipse has foci at and the constant sum equals 10.

1. What is ? What is ?
2. What is ?
3. Write the equation in standard form.
4. State the coordinates of the vertices, co-vertices, and foci.

Check-In Answer

Given: foci at so ; constant sum so

Equation:

From Geometry to Algebra

We have established the geometry. Now we derive the equation.

Setup:

• Center at origin; foci at and
• Generic point on the ellipse

The locus condition becomes:

Goal: eliminate both square roots and arrive at the standard form.

Stages 1–2: Isolate One Radical and Square

Stage 1 — Isolate the left radical:

Stage 2 — Square both sides (first squaring):

M5 Alert: Squaring once is not enough — a square root still appears on the right side. We will need a second squaring after simplification.

Stage 3: Simplify After First Squaring

Expand and then cancel , , and :

Cancel matching terms from both sides:

Rearrange and divide by :

Stage 4: Isolate Remaining Radical

This is now a single radical. Square both sides (second squaring):

Stages 5–6: Cancel, Substitute, and Arrive at Standard Form

Result: The Standard Form

This is the standard equation of an ellipse centered at the origin with:

• Major axis along the x-axis (if )
• Semi-major axis length ; semi-minor axis length
• Foci at where

Special case: if then and the equation becomes — a circle.

Verification and Reference

Verify: does satisfy ?

Does satisfy it?

Guided Practice

An ellipse has foci at and passes through the point .

1. Identify , then find from the given point.
2. Use to find .
3. Write the standard equation.
4. State all key features.

Guided Practice Answer

Step 1: Foci at so . Point is on the minor axis so .

Step 2:

Step 3:

Step 4:

Practice Set

1. For : find , , , the vertices, co-vertices, and foci.

2. An ellipse has vertices at and foci at . Write its equation.

3. An ellipse has foci at and major axis of length 10. Write the equation and state the orientation of the major axis.

Summary

Key results from this lesson:

1. An ellipse is the locus where ; this gives the standard form
2. The parameters satisfy (derived geometrically at the co-vertex)
3. The major axis runs along the direction with the larger denominator
4. The derivation requires two squarings — one radical at a time
5. A circle is the special case (foci coincide, )

Watch-out table:

Coming Up: Deck 2 — Hyperbola and Eccentricity

In Deck 2 we ask a different question:

What if we use the difference of distances instead of the sum?

This produces a hyperbola — a curve with two branches that extend to infinity.

We will also introduce eccentricity, the single number that classifies all conic sections.