Chapter 10 · Introduction

Chapter 10 — Conic Sections

Slice a cone with a flat plane and four famous curves appear — the circle, the parabola, the ellipse and the hyperbola. Each has a simple distance rule, and each rule becomes a short equation.

Around $200$ BCE the Greek geometer Apollonius of Perga took a cone, cut it with a flat plane at different angles and studied the edges of the cuts. He found four kinds of curve and gave them the names we still use: the ellipse, the parabola and the hyperbola, together with the familiar circle. For eighteen centuries these "conic sections" were admired as pure geometry. Then, in $1609$, Johannes Kepler showed that Mars moves round the Sun not in a circle but in an ellipse with the Sun at one special point, a focus. Every planet, moon and artificial satellite follows the same rule. When Chandrayaan-3 left Sriharikota in July $2023$ it was first placed in an elliptical orbit that came within about $170\ \text{km}$ of the Earth and swung out to about $36\,500\ \text{km}$; ISRO then fired its engine near the closest point again and again to stretch the ellipse until it reached the Moon.

The other conics are just as practical. The large dish antennas of ISRO's deep-space network near Bengaluru are parabolas turned about their axes: every signal arriving parallel to the axis bounces off the dish to a single point, the focus, where the receiver sits — and car headlamps use the same property in reverse. Hyperbolas appear wherever a difference of distances is fixed: two radio stations sending a pulse at the same instant let a ship measure the difference of its distances from them, which places it on a hyperbola. In this chapter you will first see how a plane slicing a double cone produces all four curves, and then, one curve at a time, turn a simple rule about distances into an equation: the circle (fixed distance from a centre), the parabola (equal distances from a focus and a line), the ellipse (fixed sum of distances from two foci) and the hyperbola (fixed difference). From each equation you will read off the foci, vertices, axes, eccentricity and latus rectum, and you will learn to build the equation from given data.

Slice a double cone→Circle: $(x-h)^2+(y-k)^2=r^2$→Parabola: $y^2=4ax$→Ellipse: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$→Hyperbola: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$
🎯 By the end of this chapter you can…
  • describe how a plane cutting a double cone gives a circle, an ellipse, a parabola, a hyperbola, or one of the degenerate cases (a point, a line, a pair of lines);
  • write the equation of a circle with centre $(h,k)$ and radius $r$, and find the centre and radius from an equation $x^2+y^2+2gx+2fy+c=0$ by completing the square;
  • derive the standard equations of the parabola, the ellipse and the hyperbola from their distance definitions;
  • read off the focus or foci, the vertices, the axes, the directrix of a parabola, the eccentricity and the length of the latus rectum from a standard equation;
  • find the equation of a circle or a conic from given data — centre, foci, vertices, a point on the curve, the eccentricity or the latus rectum — and use conics to model antennas, arches and orbits.
Teaser — One Focus, One Line, One Number

Fix a point $S$ (the focus) and a line (the directrix). The curve is made of all points $P$ whose distance from $S$ is $e$ times their distance from the line: $PS=e\cdot PM$. Press Play to slide $e$ from $0.2$ up to $1.8$. While $e\lt1$ the curve closes up into an ellipse; at exactly $e=1$ it opens into a parabola; for $e\gt1$ it splits into the two branches of a hyperbola. The point $P$ runs round the curve, and the readouts show that $PS\div PM$ never changes.

0.6
curve—
$PS$—
$PM$—
$PS\div PM$—

Sections in this chapter

10.1

Sections of a Cone and the Circle

How a plane slicing a double cone gives each conic and the degenerate cases; the circle with centre $(h,k)$; the general equation and completing the square; circles through given points.

$(x-h)^2+(y-k)^2=r^2$
Open section →10.2

The Parabola

Focus and directrix, the four standard equations, the axis, vertex and latus rectum, equations from given data, and the reflecting property behind dish antennas.

$y^2=4ax,\ \ \text{LR}=4a$
Open section →10.3

The Ellipse

A constant sum of distances from two foci, major and minor axes, $c^2=a^2-b^2$, eccentricity, latus rectum, equations from given data, and satellite orbits.

$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1,\ \ e=\dfrac ca\lt1$
Open section →10.4

The Hyperbola

A constant difference of distances, transverse and conjugate axes, $c^2=a^2+b^2$, eccentricity greater than $1$, latus rectum, and telling the standard forms apart.

$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1,\ \ e=\dfrac ca\gt1$
Open section →
✅ Before you start
From Class 10 Mathematics you need the distance formula $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ and the mid-point formula (Chapter 7, Coordinate Geometry), and confidence with squaring both sides of an equation and with expanding $(a\pm b)^2$. Completing the square, which you met while solving quadratic equations (Class 10, Chapter 4), is the key tool for circles. The idea of a graph as the set of all points that satisfy an equation, from Section 2.3, is used throughout, and the angle measures of Section 3.1 appear when we slice the cone. Chapter 9 of this course (Straight Lines) supplies the equation of a line such as $x=-a$ for a directrix; nothing else from it is needed.
← 9.3 Distance of a Point from a Line10.1 Sections of a Cone and the Circle →