Chapter 12
Conic Sections
Shine a flashlight straight at a wall and the bright patch is a circle. Tilt it and the circle stretches into an oval. Tilt further and the patch stops closing up at all. Nothing about the light changed, only the angle at which the wall cuts across the cone. The same curves appear in the path of a thrown ball and in the orbit of a planet. They are not a random collection: each comes from a rule about distance, and the surprise is how such a rule becomes an equation you can graph.
What You'll Explore
4 lessons.
- Parabolas
You have graphed parabolas since your first quadratic, without ever being told what separates that curve from any other bowl shape. Here it is described by distances alone, and that same description is what a satellite dish and a car headlight are built around.
- Circles
Circles are the easiest of these curves to state and the easiest to disguise. Multiply out the equation, shuffle the terms, and neither the center nor the radius is anywhere in sight. Given only that jumble, can you always recover the circle it came from, and is a circle always what you get back?
- Ellipses
Push two pins into a board, loop a string around them, and drag a pencil around the inside of the loop with the string pulled taut. The curve that comes out is an oval rather than a circle. What is the pencil obeying at every moment, and what does moving the pins change about the shape?
- Hyperbolas
Every curve so far has been a single connected piece. A hyperbola arrives in two, facing away from each other, and each branch runs off toward a pair of straight lines it never quite reaches. Where does the second branch come from, and how can a curve approach a line forever without meeting it?