A sketch of the locus of a complex number 
is shown above, determine the maximum value of 
correct to two decimal places where 
Draw tangent lines from the origin to the circle.
Remember tangent lines are perpendicular to the radii
The maximum argument is this angle
I am going to find the angle in two sections
From the diagram the radius of the circle is 
and the centre is 
. Hence the distance from the origin to the centre is 
.
Hence maximum 
Let 
be the region of the complex plane where the inequalities 
and 
hold simultaneously.
First find the Cartesian equations.
Finding the symbols The  is found underMenu – Number – Complex Number Tools – Complex Conjugate  |
Second, sketch each of the functions.
The section that is shaded twice is our region.
Determine the minimum value of 
in .
We can find the point of intersection between the circle and the line.
Or if you want exact values
Use the Solve Systems of Equations tool – Menu – Algebra – Solve Systems of Equations – Solve Systems of Equations.
I found this question on the madasmaths site – his resources are fabulous.
I am not sure I would have been able to do this in an exam.
I have split my solution into 6 images and there is a pdf version at the bottom
PDF version of my solution
Filed under Complex Numbers, Sketching Complex Regions
