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Conic Sections

The Hook

Slicing the cone.

What is it?

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

Time Travel (History)

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

Cheat Code

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

Practice This Now!

The Field Guide

In 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.

In The Real World: 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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Booby Trap!

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

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

Training Drills

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