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Conic Sections
Introduction
Sections of a Cone
Circle
Practice: Circle Equations
Parabola
Practice: Parabola Equations
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Practice: Ellipse Equations
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Practice: Hyperbola Equations
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Chapter Review: Mixed Practice
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Glossary

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Conic SectionsIntroduction

Reading time: ~15 min

It is eight o'clock on a summer evening and the power has gone again. Aman finds the emergency torch, and his sister Riya shines it at the bedroom wall. Held straight, it makes a neat round spot of light. Riya tips the torch upwards and the spot stretches into an oval. She tips it a little more, and suddenly the patch no longer closes at all: it streams up the wall and out of sight.

Those shapes are not random. They are the curves this whole chapter is about, and you are about to make all of them yourself.

You are holding the torch. On the left is the side view: the torch, its cone of light and the wall. On the right is the wall itself, as Riya sees it.

Tilt the torch and complete all four missions.

Tilt of the torch: ${t}°

Every one of these patches is the place where a cone of light meets a flat plane, the wall. Slice a cone with a plane and you always get one of these curves: a , an ellipse, a parabola or a hyperbola. That is why they are called conic sections, or simply conics.

In the last chapter you studied the equations of straight . Now you will meet these four curves, find their equations, and see why they turn up everywhere: in the orbits of planets, in telescopes and satellite dishes, in car headlights and in the arches of bridges.

Two Thousand Years of Conics

People have studied these curves for well over two thousand years. Around 300 BCE Euclid collected the geometry of his time into one great book, building everything from a handful of basic . About a century later, around 200 BCE, Apollonius of Perga wrote a work in eight books called Conics. He showed that a single double cone, cut at different angles, gives every one of these curves, and he gave the parabola and the hyperbola the names we still use.

For centuries conics were pure geometry, studied for their own beauty, as Archimedes did too. Then in 1609 Kepler found that every planet moves round the Sun along an ellipse. In 1637 René Descartes published La Géométrie, which describes curves by equations in x and y — the coordinate geometry you use today. Pierre de Fermat had found the same idea on his own, but his book on it was printed only in 1679, after his death, so the method carries Descartes' name. In 1750 Gabriel Cramer wrote the equation of a circle using two axes, close to the form you are about to learn.

Drag these milestones into the order in which they happened, earliest at the top.

Descartes publishes La Géométrie
Euclid builds geometry from a few axioms
Cramer writes the equation of a circle using two axes
Kepler finds that planets move in ellipses
Fermat's book on curves is finally printed
Apollonius writes Conics and names the parabola and hyperbola

Today conics steer spacecraft, focus telescopes and shape the reflector behind every car headlight. In this chapter you will see where each curve comes from, write its equation, and read off everything about it — centre, foci, directrix, eccentricity, latus rectum — straight from that equation.

Sina