# Thread: Complex Numbers (Loci)

1. ## Complex Numbers (Loci)

Hello!

Q: The complex number w is represented by the point of intersection of the loci |z -8 -6i| = 5 and arg(z -4 -3i) = 2 $\alpha$, where $\alpha$ = $\arctan$ (0.75). Find w, in the form x + iy, giving the exact decimal values of x and y.

Thank you for helping!

2. Originally Posted by Tangera
Hello!

Q: The complex number w is represented by the point of intersection of the loci |z -8 -6i| = 5 and arg(z -4 -3i) = 2 $\alpha$, where $\alpha$ = $\arctan$ (0.75). Find w, in the form x + iy, giving the exact decimal values of x and y.

Thank you for helping!
There is probably some simple geometric way of doing this - it can't be coincidence that the ray $\text{arg} (z - 4 -3i) = \alpha$ passes through the centre of the circle $|z -8 -6i| = 5$ and that the terminus of the ray lies on this circle.

But a blunt apporach is:

Get the Cartesian equation of $|z -8 -6i| = 5$, easy if you recognise it as a circle of radius 5 and centre at (8, 6).

Get the equation of the line that the ray $\text{arg} (z - 4 -3i) = 2 \alpha$ lies on. Note that the gradient of this line is $\tan 2 \alpha = \frac{2 \tan \alpha}{1 - \tan^2 \alpha}$ and it passes through the point (4, 3).

Get the intersection of the circle and the line. The solution you want to keep is the one such that y > 3 (a diagram makes this clear).