# A challenging complex number problem with solution

By | September 27, 2011

This complex number problem was selected on the basis of its uniqueness in terms of phrasing things within the Argand diagram/locus context. While my proposed solutions might be short, bear in mind this question truly demands/challenges the student to think unconventionally in order to formulate a viable solving approach.

Problem

A complex number $z=x+iy$ is represented by the point $P$ in an Argand diagram. If the complex number $w$ where $w = \frac{z-8i}{z+6}, (z\neq-6)$ has its real part zero, show that the locus of $P$ in the Argand diagram is a circle and find the radius and the coordinates of the centre of this circle. If, however, $w$ is real, find the locus of $P$ in this case.

Solution

The author of this post is Mr. Frederick Koh. He is a teacher residing in Singapore who specialises in teaching the A level maths curriculum. He has accumulated more than a decade of tutoring experience and loves to share his passion for mathematics on his personal site www.whitegroupmaths.com.

If you love this problem, I’m sure you will also enjoy the two other challenging problems shared by Mr. Koh in this site: