Given that |z – 2i| < 4, illustrate the locus of the point representing the complex number z in an Argand diagram. Hence find the greatest and least possible values of |z – 3 + 4i|, given that |z – 2i|< 4.

Source: 2003 GCE A-Level Mathematics

How can we approach the question?

z is a set of points when joined to point 2i create lines of length less than or equal to 4 units. Think of the circle of radius 4 units, with centre at 2i joining to points on the circumference. z is the set of points within the circle, including the circumference.

complex01

|z – 3 + 4i| = |z – (3 – 4i)| is the distance between the point z and the point (3, -4). For the greatest value and least distance, we need to identify the correct z from the set of points. (3, -4) is a point outside of the circle. The diameter of the circle is the longest chord in any circle. Hence, the greatest and least distance are by the points (z1 and z2 respectively as shown below) on the circumference joining the centre to the point (3, -4).

complex02

complex03

For more hands-on practice on how to draw loci of points in an Argand diagram, as well as understanding complex numbers, join our A-Level Mathematics Tuition Programme.

 

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