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 opens into a curve running off the edges. Swap the flashlight for a real double cone, two cones joined tip to tip like an hourglass, and a single slice through both halves traces two curves instead of one. The same curves trace a thrown ball's path and a planet's orbit. Each comes from a plain rule about distance, and the surprise is how that rule becomes an equation you can graph.
What You'll Explore
4 lessons.
Some lessons are marked Advanced because they go beyond core Algebra II. Skip them or explore them if you are curious.
- Parabolas Advanced. This lesson goes beyond core Algebra II. You can skip it.
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 Advanced. This lesson goes beyond core Algebra II. You can skip it.
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 Advanced. This lesson goes beyond core Algebra II. You can skip it.
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?