6Quadratic forms and conics

IA Vectors and Matrices

6.2 Focus-directrix property

Conic sections can be defined in a different way, in terms of

Definition

(Conic sections)

.

The eccentricity and scale are properties of a conic

section that satisfy the following:

Let the foci of a conic section be (±ae, 0) and the directrices be x = ±a/e.

A conic section is the set of points whose distance from focus is

e×

distance

from directrix which is closer to that of focus (unless

e

= 1, where we take the

distance to the other directrix).

Now consider the different cases of e:

(i) e < 1. By definition,

x

y

O

x = a/e

ae

(x, y)

p

(x − ae)

2

+ y

2

= e

a

e

− x

x

2

a

2

+

y

2

a

2

(1 − e

2

)

= 1

Which is an ellipse with semi-major axis

a

and semi-minor axis

a

√

1 − e

2

.

(if e = 0, then we have a circle)

(ii) e > 1. So

x

y

O

x = a/e

ae

(x, y)

p

(x − ae)

2

+ y

2

= e

x −

a

e

x

2

a

2

−

y

2

a

2

(e

2

− 1)

= 1

and we have a hyperbola.

(iii) e = 1: Then

x

y

O

x = a

a

(x, y)

p

(x − a)

2

+ y

2

= (x + 1)

y

2

= 4ax

and we have a parabola.

Conics also work in polar coordinates. We introduce a new parameter

l

such

that l/e is the distance from the focus to the directrix. So

l = a|1 − e

2

|.

We use polar coordinates (

r, θ

) centered on a focus. So the focus-directrix

property is

r = e

l

e

− r cos θ

r =

l

1 + e cos θ

We see that

r → ∞

if

θ → cos

−1

(

−

1

/e

), which is only possible if

e ≥

1, i.e.

hyperbola or parabola. But ellipses have e < 1. So r is bounded, as expected.