Lesson 15.1

Introduction to Conic Sections

Visualizing circles, ellipses, parabolas, and hyperbolas as slices of a double cone.

Introduction

Conic sections are curves formed by the intersection of a plane and a double cone. Depending on the angle of the cut, we get four distinct shapes: circles, ellipses, parabolas, and hyperbolas.

1

Prerequisite Connection

You understand coordinate geometry and can work with second-degree equations.

2

Today's Increment

We introduce all four conics as geometric loci and see how they arise from slicing a cone.

3

Why This Matters

Conics appear everywhere: planetary orbits (ellipses), satellite dishes (parabolas), and navigation systems (hyperbolas).

The Four Conic Sections

How Conics Are Formed

Imagine a double cone (two cones tip-to-tip). A plane slicing through it creates:

  • • Circle: Plane perpendicular to the cone's axis
  • • Ellipse: Plane tilted, but doesn't hit the base
  • • Parabola: Plane parallel to the cone's side
  • • Hyperbola: Plane cuts through both halves of the double cone

Circle

All points equidistant from center

Ellipse

Sum of distances to two foci is constant

Parabola

Distance to focus = distance to directrix

Hyperbola

Difference of distances to foci is constant

The General Second-Degree Equation

Every conic section can be written in this form. The values of , , and determine which conic it is.

Discriminant Test (when B = 0)

  • • : Circle
  • • and same sign but : Ellipse
  • • or (but not both): Parabola
  • • and opposite signs: Hyperbola

Worked Examples

Example 1: Identifying a Conic from Its Equation

Identify the conic:

Step 1: Rewrite in standard form

Step 2: Check coefficients

Both terms are positive and added, with different denominators (9 ≠ 4).

Solution

This is an ellipse centered at the origin with and .

Example 2: Recognizing a Hyperbola

Identify the conic:

Step 1: Rewrite in standard form

Step 2: Check the sign

One term is subtracted. The term is positive, so the hyperbola opens left-right.

Solution

This is a hyperbola with and , opening horizontally.

Example 3: Identifying a Parabola

Identify the conic:

Step 1: Analyze the equation

Only one variable is squared (). This is the hallmark of a parabola.

Step 2: Find the parameter

The focus is at .

Solution

This is a parabola opening upward with vertex at origin and focus at .

Common Pitfalls

Confusing ellipse and hyperbola

Ellipse: between squared terms. Hyperbola: between them.

Forgetting that circles are special ellipses

A circle is an ellipse where . If coefficients are equal, it's a circle!

Not dividing to get = 1

Standard form requires the right side to equal 1. Don't forget to divide the entire equation.

Real-World Application

Planetary Orbits

Johannes Kepler discovered that planets orbit the sun in ellipses, not circles. The sun sits at one focus of the ellipse.

Comets often travel in highly eccentric ellipses or even parabolic/hyperbolic paths—the latter means they only pass through once and never return!

Practice Quiz

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