## A tricky complex numbers problem

The question is as follows:

If and and , find .

So, let’s think about the information given and what we are trying to find. We want to find the complex number which satisfies this slightly strange set of constraints, and the constraints are given in terms of and . So, by the looks of things, the answer will depend on and so the final expression should be a function of .

Now let’s explore the constraints. In fact, let’s simply take and as two complex numbers, but importantly two numbers which differ only by a real number , so wherever they lie in the complex plane, they have the same imaginary part and differ only by an real part.

Now, the constraints are about the arguments of the two complex numbers. It doesn’t tell us anything about the magnitude of the numbers, so all the information tells us is the direction are in relation to the origin.…