Lesson 1 of 12 · 7 min

Sections of a cone

NCERT §10.1, §10.2, §10.2.1

At the science club's terrace night, Meera shines a torch at the wall. Pointed straight, the patch of light is round; tilted, it stretches into an oval; tilted further, one edge runs off and never closes. Why does one beam give three shapes?

The story this chapter follows: The science club's terrace night

Imagine Meera and her school science club spending an evening on the gridded school terrace, measured in metres with a water tap at the origin. A torch beam cut by a wall, a round flower bed, a torch reflector, a string-and-peg oval garden, two listeners timing a clap and a solar cooker's mirror each turn out to be one of the four conic sections.

The lesson in notes

In short

Circles, ellipses, parabolas and hyperbolas are called conic sections, or conics, because each one is the curve where a flat plane meets a double napped right circular cone. The names parabola and hyperbola go back to Apollonius.

Build the cone from two lines: a fixed vertical line l and a line m that crosses it at a point V at a fixed angle α. Spin m about l, keeping α unchanged, and m sweeps out a double napped right circular hollow cone that runs on without end both ways.

V is the vertex, l is the axis and the spinning line m is a generator. The vertex splits the cone into two pieces, the upper and lower nappes.

Let β be the angle the cutting plane makes with the vertical axis. When the plane does not pass through V, the curve depends only on how β compares with α.

β = 90° (plane at right angles to the axis): a circle. α < β < 90°: an ellipse. β = α (plane parallel to a generator): a parabola. In these three cases the plane cuts across one nappe only.

0 ≤ β < α: the plane is steeper than the generators, cuts through both nappes, and the curve is a hyperbola, which therefore has two separate branches.

These curves turn up in planetary orbits, telescope and antenna design, and the reflectors of torches and vehicle headlights.