Hyperbola and Eccentricity

HSG.GPE.A.3 — Lesson 2 of 2

Expressing Geometric Properties with Equations

Learning Objectives

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

3. Identify key features of ellipses and hyperbolas from their equations: center, vertices, foci, axes, and asymptotes
4. Compute eccentricity and explain how it describes shape
5. Distinguish between ellipses and hyperbolas based on equations and geometric definitions

Hook: Planets Orbit in Ellipses. What About Comets?

Most comets orbit the Sun in highly elongated ellipses (eccentricity close to 1).

But some comets approach the Sun on a hyperbolic trajectory — they pass through the solar system once and never return.

The difference between these two fates:

• Elliptical orbit: eccentricity — the comet is bound to the Sun
• Hyperbolic trajectory: eccentricity — the comet escapes forever

The same mathematics governs both. The only difference is one sign.

The Hyperbola: Difference Instead of Sum

An ellipse uses the sum of distances:

A hyperbola uses the difference:

Why for a Hyperbola

Triangle inequality: The difference of distances from any point to two fixed points and is at most the distance :

For the hyperbola condition to be achievable, we need:

Intuition: The constant difference must be strictly less than the full separation between foci — otherwise no point can satisfy the condition.

Derivation: Same Technique, One Sign Change

The derivation mirrors the ellipse — apply the same isolate-and-square technique.

Start:

After two squarings (same algebra as the ellipse, with subtraction):

But now , so . Define .

Substituting and dividing by :

M1 Key contrast: Ellipse: . Hyperbola: .

Two Branches, Not a Closed Curve

M3 Watch-out: A hyperbola is NOT a closed curve. It has two separate branches that extend to infinity.

Why two branches?

So (left branch) or (right branch). No points exist between .

The Hyperbola: Both Branches and Asymptotes

Asymptotes:

For the hyperbola , solve for :

As : , so

The hyperbola approaches — but never reaches — the lines .

Vertical Transverse Axis:

When the minus sign is under instead of , the transverse axis is vertical.

M4 Watch-out: The asymptotes pass through the center, not the vertices. The axes of symmetry are still the x- and y-axes — not the asymptotes.

Check-In

For the hyperbola :

1. Identify , , and .
2. State the vertices and foci.
3. Write the equations of the asymptotes.
4. Is the eccentricity greater than or less than 1? Compute it.

Check-In Answer

For :

; ;

From Features to Comparison

Now let us compare ellipses and hyperbolas side by side — and then unify them through eccentricity.

Ellipse feature summary:

Hyperbola Feature Summary

M4 note: Asymptotes () are diagonal lines through the center. The axes of symmetry are the coordinate axes — distinct from the asymptotes.

Side-by-Side Comparison

Worked Example: Ellipse

Given:

Worked Example: Hyperbola

Given:

Eccentricity: The Shape Spectrum

Real-World Eccentricities

Kepler's First Law: Every planetary orbit is an ellipse with the Sun at one focus.

The Complete Comparison Table

Guided Practice

For the hyperbola :

1. Identify , , and .
2. State the vertices and foci.
3. Write the equations of the asymptotes.
4. Compute the eccentricity.
5. Sketch the hyperbola (mark vertices, foci, central rectangle, and asymptotes).

Guided Practice Answer

For :

; ;

Note: , , is a Pythagorean triple — a clean check!

Summary

Key results from Deck 2:

1. A hyperbola uses ; this gives the standard form
2. For hyperbolas: and (reversed from ellipses)
3. Asymptotes are approached but never reached
4. Eccentricity classifies all conic sections: ellipse, parabola, hyperbola
5. The only difference between ellipse and hyperbola equations is the sign between the two fractions

Watch-out table:

What Comes Next

HSG.GPE.B — Using Coordinates to Prove Geometric Theorems

In the next cluster we use the coordinate tools we have built — distance, midpoint, slope — to prove geometric theorems algebraically.

Conic sections in HSG.GPE.A gave us the foundation:

• Any geometric locus condition can be expressed as an algebraic equation
• The equation encodes all geometric properties

That same principle drives the proofs ahead.

The conic section family is complete: