Lesson 15.4

The Hyperbola: Asymptotes and Vertices

Understanding the difference between addition and subtraction in conic standard forms.

Introduction

A hyperbola is the set of all points where the difference of distances to two foci is constant. Unlike the ellipse, it has two separate branches.

1

Prerequisite Connection

You understand ellipses and the concept of foci and eccentricity.

2

Today's Increment

We learn hyperbola standard forms, find asymptotes, vertices, and foci.

3

Why This Matters

Hyperbolas are used in navigation (LORAN), cooling towers, and the paths of some comets.

Asymptotes and Vertices

Horizontal Transverse Axis

Opens left and right

Vertical Transverse Axis

Opens up and down

Key Relationship (Different from Ellipse!)

Note: For hyperbolas we add, not subtract

Asymptotes

Terminology

  • • Transverse axis: The axis containing the vertices (length = )
  • • Conjugate axis: Perpendicular to transverse (length = )
  • • Vertices: Points on the hyperbola closest to center ( on transverse axis)
  • • Foci: On transverse axis, units from center

Interactive: Adjust a and b

a = 3b = 2c ≈ 3.61asymptote: y = ±0.67x

Worked Examples

Example 1: Finding Components

Find the vertices, foci, and asymptotes of .

Step 1: Identify a and b

,

Step 2: Find c

Solution

Vertices:
Foci:
Asymptotes:

Example 2: Vertical Hyperbola

Graph and analyze .

Step 1: Identify orientation

is positive, so transverse axis is vertical. Opens up and down.

Step 2: Find values

, ,

Solution

Vertices:
Foci:
Asymptotes:

Example 3: Translated Hyperbola

Find the center, vertices, and asymptotes of .

Step 1: Identify center

Center:

Step 2: Find a and b

, . Horizontal transverse axis.

Solution

Center:
Vertices: and
Asymptotes:

Common Pitfalls

Using subtraction for c²

For hyperbolas: (add!). Ellipses subtract.

Confusing which term is "a"

is always under the positive term, regardless of which variable it's with.

Wrong asymptote slope

For horizontal: . For vertical: . The order matters!

Real-World Application

LORAN Navigation

LORAN (Long Range Navigation) uses the time difference of radio signals from two stations. The set of positions with a constant time difference forms a hyperbola!

By measuring differences from multiple pairs of stations, ships and aircraft can pinpoint their exact location—a real-world application of the hyperbola's definition.

Practice Quiz

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