**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.

__<__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.

|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 (z_{1} and z_{2} respectively as shown below) on the circumference joining the centre to the point (3, -4).

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