Lesson 15.5

Rotation of Conic Sections

Understanding the xy term and how to rotate coordinate axes to eliminate it.

Introduction

When a conic section is rotated, an term appears in the equation. To analyze such conics, we rotate the coordinate system to eliminate this term.

1

Prerequisite Connection

You can graph and analyze all four conic sections in standard position.

2

Today's Increment

We learn the rotation formulas, the discriminant test, and how to identify rotated conics.

3

Why This Matters

Real-world conics are rarely axis-aligned. Learning rotation helps analyze any orientation.

The xy Term and Rotation

General Second-Degree Equation

When , the conic is rotated

Rotation Formulas

These convert from rotated (x', y') to original (x, y) coordinates

Rotation Angle to Eliminate xy

or equivalently:

Discriminant Test (Identifies Conic Type)

  • • : Ellipse (or circle if and )
  • • : Parabola
  • • : Hyperbola

Interactive: Rotating a Conic

Gray dashed: original ellipse | Blue: rotated by 45°

Key Insight

The discriminant is invariant under rotation. No matter how you rotate the axes, this value stays the same—so it always correctly identifies the conic type.

Worked Examples

Example 1: Identifying a Rotated Conic

Identify the conic:

Step 1: Identify A, B, C

, ,

Step 2: Calculate discriminant

Solution

Since , this is a hyperbola.

Example 2: Finding the Rotation Angle

Find the angle needed to eliminate the xy term in .

Step 1: Identify coefficients

, ,

Step 2: Apply rotation formula

Step 3: Solve for θ

means , so .

Solution

Rotate the axes by 45° to eliminate the xy term.

Example 3: Classifying Multiple Conics

Classify each conic using the discriminant:

(a)

→ Hyperbola

(b)

→ Parabola

(c)

→ Ellipse

Common Pitfalls

Forgetting the coefficient of xy

In , the coefficient , not 1. Read carefully!

Using θ instead of 2θ

The formula gives . Don't forget to divide by 2 when finding θ.

Confusing discriminant signs

Negative discriminant → Ellipse (closed curve). Positive → Hyperbola (two branches).

Real-World Application

Computer Graphics and Image Processing

When cameras capture images at an angle, circles become ellipses. Understanding rotation helps correct these distortions and recognize shapes regardless of orientation.

The rotation formulas are used extensively in 3D graphics to transform objects and in machine learning for pattern recognition—the math of alignment!

Practice Quiz

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