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Conic Sections

Overview

Slicing the cone.

What is it?

Curves obtained by slicing a cone: Circle, Ellipse, Parabola, Hyperbola.

History

Studied by Greeks 2000 years ago, but became vital later for describing planetary orbits (Kepler).

Key Idea

Plus sign = Ellipse (Planets). Minus sign = Hyperbola (Comets).

Practice This Topic

Concept Guide

Plain English: Imagine a traffic cone. If you slice it straight, you see a Circle. Slice it on a slant, an Ellipse (oval). Slice it parallel to the side, a Parabola (U-shape). These 4 shapes describe gravity.

Real-world example: Satellite Dishes. They are parabolic so all signals bounce to a single focus point.

How to do it

  1. Look at the squared terms.
  2. Only one squared? Parabola.
  3. Both squared, same number? Circle.
  4. Both squared, different numbers, Plus sign? Ellipse.
  5. Both squared, Minus sign? Hyperbola.
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Common Pitfall

Mixing up the equations. Ellipse has a '+' between x² and y². Hyperbola has a '-'.

Word Problem
"A whispering gallery is shaped so you can hear a pin drop from across the room. What shape is it?"
Reasoning: An Ellipse. Sound from one focus point bounces perfectly to the other focus point.

Practice Examples

Circle
x² + y² = r²
Round.
Ellipse
x²/a² + y²/b² = 1
Oval.