vLet the relation of knowledge to real life be very visible to your pupils

and let them understand how by knowledge the world could be

transformed. – BERTRAND RUSSELL v

11.1 Introduction

In the preceding Chapter 10, we have studied various forms

of the equations of a line. In this Chapter, we shall study

about some other curves, viz., circles, ellipses, parabolas

and hyperbolas. The names parabola and hyperbola are

given by Apollonius. These curves are in fact, known as

conic sections or more commonly conics because they

can be obtained as intersections of a plane with a double

napped right circular cone. These curves have a very wide

range of applications in fields such as planetary motion,

design of telescopes and antennas, reflectors in flashlights

and automobile headlights, etc. Now, in the subsequent sections we will see how the

intersection of a plane with a double napped right circular cone

results in different types of curves.

11.2 Sections of a Cone

Let l be a fixed vertical line and m be another line intersecting it at

a fixed point V and inclined to it at an angle α (Fig11.1).

Suppose we rotate the line m around the line l in such a way

that the angle α remains constant. Then the surface generated is

a double-napped right circular hollow cone herein after referred as

Apollonius

(262 B.C. -190 B.C.)

11Chapter

Fig 11. 1

CONIC SECTIONS

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CONIC SECTIONS 237

Fig 11. 2

Fig 11. 3

cone and extending indefinitely far in both directions (Fig11.2).

The point V is called the vertex; the line l is the axis of the cone. The rotating line

m is called a generator of the cone. The vertex separates the cone into two parts

called nappes.

If we take the intersection of a plane with a cone, the section so obtained is called

a conic section. Thus, conic sections are the curves obtained by intersecting a right

circular cone by a plane.

We obtain different kinds of conic sections depending on the position of the

intersecting plane with respect to the cone and by the angle made by it with the vertical

axis of the cone. Let β be the angle made by the intersecting plane with the vertical

axis of the cone (Fig11.3).

The intersection of the plane with the cone can take place either at the vertex of

the cone or at any other part of the nappe either below or above the vertex.

11.2.1 Circle, ellipse, parabola and hyperbola When the plane cuts the nappe (other

than the vertex) of the cone, we have the following situations:

(a) When β = 90

o

, the section is a circle (Fig11.4).

(b) When α < β < 90

o

, the section is an ellipse (Fig11.5).

(c) When β = α; the section is a parabola (Fig11.6).

(In each of the above three situations, the plane cuts entirely across one nappe of

the cone).

(d) When 0 ≤ β < α; the plane cuts through both the nappes and the curves of

intersection is a hyperbola (Fig11.7).

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238 MATHEMATICS

Fig 11. 4

11.2.2 Degenerated conic sections

When the plane cuts at the vertex of the cone, we have the following different cases:

(a) When α < β ≤ 90

o

, then the section is a point (Fig11.8).

(b) When β = α, the plane contains a generator of the cone and the section is a

straight line (Fig11.9).

It is the degenerated case of a parabola.

(c) When 0 ≤ β < α, the section is a pair of intersecting straight lines (Fig11.10). It is

the degenerated case of a hyperbola.

Fig 11. 6

Fig 11. 7

Fig 11. 5

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CONIC SECTIONS 239

In the following sections, we shall obtain the equations of each of these conic

sections in standard form by defining them based on geometric properties.

Fig 11. 8

Fig 11. 9

Fig 1

1. 10

11.3 Circle

Definition 1 A circle is the set of all points in a plane that are equidistant from a fixed

point in the plane.

The fixed point is called the centre of the circle and the distance from the centre

to a point on the circle is called the radius of the circle (Fig 11.11).

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240 MATHEMATICS

The equation of the circle is simplest if the centre of the circle is at the origin.

However, we derive below the equation of the circle with a given centre and radius

(Fig 11.12).

Given C (h, k) be the centre and r the radius of circle. Let P(x, y) be any point on

the circle (Fig11.12). Then, by the definition, | CP | = r . By the distance formula,

we have

2 2

( ) ( )

x – h y – k r

+ =

i.e. (x – h)

2

+ (y – k)

2

= r

2

This is the required equation of the circle with centre at (h,k) and radius r .

Example 1 Find an equation of the circle with centre at (0,0) and radius r.

Solution Here h = k = 0. Therefore, the equation of the circle is x

2

+ y

2

= r

2

.

Example 2 Find the equation of the circle with centre (–3, 2) and radius 4.

Solution Here h = –3, k = 2 and r = 4. Therefore, the equation of the required circle is

(x + 3)

2

+ (y –2)

2

= 16

Example 3 Find the centre and the radius of the circle x

2

+ y

2

+ 8x + 10y – 8 = 0

Solution The given equation is

(x

2

+ 8x) + (y

2

+ 10y) = 8

Now, completing the squares within the parenthesis, we get

(x

2

+ 8x + 16) + (y

2

+ 10y + 25) = 8 + 16 + 25

i.e. (x + 4)

2

+ (y + 5)

2

= 49

i.e. {

x – (– 4)}

2

+ {y – (–5)}

2

= 7

2

Therefore, the given circle has centre at (– 4, –5) and radius 7.

Fig 11. 11

Fig 11. 12

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