“Greatest and least possible values of |z|” type questions

Example

The point P represents a complex number z on an Argand diagram. Given that |z+1−i|=1:

a. find a Cartesian equation for the locus of P

b. sketch the locus of P on an Argand diagram

c. find the greatest and least possible values of |z|

d. find the greatest and least possible values of |z-1|

How to do it

The key here is to do a diagram and to look for your answers from the diagram, rather than from algebra.

So start with part b: the locus is the circle of all the points that have a distance of 1 from the point -1+i

Draw that circle.

Use your knowledge that a circle with centre (a,b) and radius r has equation

(x-a)2 + (y-b)2 = r2

to find the “Cartesian equation” (i.e. equation in x and y) asked for in part a.

Then for part c, look at your diagram and think: at what points would z have its greatest and least possible values of |z|, in other words of its distance from the origin?

That would be when z is on the P-circle you’ve drawn |z+1−i|=1 and simultaneously on the biggest possible circle with centre the origin (for the greatest value of |z|) and the smallest possible ditto circle (for the least value of |z|)

You find those two points by drawing a line from the origin through the centre of the P-circle (i.e. through −1+i) and seeing the two points where the line cuts the P-circle – the furthest-away point and the nearest-to-the-origin point.

Work out where those two points are by using geometry and trig.

For example, the furthest-away point is a distance √2+1 from the origin:

√2 to the centre of the P-circle, and then another 1 from the centre of the P-circle to the circumference – and in a direction 3π/4 anticlockwise from the positive real axis.

In this question, they only want the distance, √2+1, but if they wanted the coordinates of the point you could find those too.

For greatest and least values of |z−1|, same thing, except it’s given by biggest possible circle with centre z=1 and smallest possible circle with centre z=1, and the points are where a line from the point z=1 (instead of the origin), through the centre of the P-circle, crosses the P-circle.