Complex numbers: Lecture 7

In this lecture you have learnt how to draw the loci in the form of \[arg(z-a)=\theta\]

Remember the description of such locus:

Let A and P be the points in the Argand diagram representing a and z respectively,

The locus is the half-line from A (excluding A) that makes an angle theta with the positive real axis.

In addition, we need to know the following geometrical knowledge:

1. Given a point A outside a circle with centre O, find a point B on the circle such that AB is longest.

Answer: B is the point such that AB passes through O.

2. Given a point A outside a circle with centre O, find a point B on the circle such that AB is shortest.

Answer: B is the point such that the extended AB passes through O.

3. Given a point A outside a circle with centre O, find a point B on the circle such that angle BAO is maximum.

Answer: B is the point such that AB is the tangent to the circle. (There are two such point B.)

This is the last lecture on complex numbers. Now you should finish the tutorial as soon as possible.