1Complex numbers

IA Vectors and Matrices

1.6 Lines and circles in C

Since complex numbers can be regarded as points on the 2D plane, we can often

use complex numbers to represent two dimensional objects.

Suppose that we want to represent a straight line through

z

0

∈ C

parallel to

w ∈ C

. The obvious way to do so is to let

z

=

z

0

+

λw

where

λ

can take any

real value. However, this is not an optimal way of doing so, since we are not

using the power of complex numbers fully. This is just the same as the vector

equation for straight lines, which you may or may not know from your A levels.

Instead, we arrange the equation to give

λ

=

z−z

0

w

. We take the complex

conjugate of this expression to obtain

¯

λ

=

¯z− ¯z

0

¯w

. The trick here is to realize that

λ is a real number. So we must have λ =

¯

λ. This means that we must have

z −z

0

w

=

¯z − ¯z

0

¯w

z ¯w − ¯zw = z

0

¯w − ¯z

0

w.

Theorem

(Equation of straight line)

.

The equation of a straight line through

z

0

and parallel to w is given by

z ¯w − ¯zw = z

0

¯w − ¯z

0

w.

The equation of a circle, on the other hand, is rather straightforward. Suppose

that we want a circle with center

c ∈ C

and radius

ρ ∈ R

+

. By definition of a

circle, a point

z

is on the circle iff its distance to

c

is

ρ

, i.e.

|z −c|

=

ρ

. Recalling

that |z|

2

= z¯z, we obtain,

|z − c| = ρ

|z − c|

2

= ρ

2

(z − c)(¯z − ¯c) = ρ

2

z¯z − ¯cz − c¯z = ρ

2

− c¯c

Theorem.

The general equation of a circle with center

c ∈ C

and radius

ρ ∈ R

+

can be given by

z¯z − ¯cz − c¯z = ρ

2

− c¯c.