Complex numbers, loci and arc

1. Dec 15, 2016

Kajan thana

1. The problem statement, all variables and given/known data

Sketch the loci, find centre point and the radius of the circle.
args((z-3i)/((z+4))=π/6

2. Relevant equations
args(x/y)=args(x)-args(y)
Circle theorem - inclined angle theorem

3. The attempt at a solution

I sketched the circle with major arc.
H=O/sinθ . H=2.5/sin(π/6)

I am stuck on finding the centre point.

2. Dec 15, 2016

Staff: Mentor

Can you find one or two points on the circle?

3. Dec 15, 2016

Ray Vickson

Write $z = x + iy$ and express the ratio $(z-3i)/(z+4)$ as $A(x,y) + i B(x,y)$. How can you get the equation of the curve in terms of the functions $A(x,y)$ and $B(x,y)$?

4. Dec 16, 2016

Kajan thana

I don't know how to change it into that form.

Last edited: Dec 16, 2016
5. Dec 16, 2016

Kajan thana

The coordinates are (0,3) and (-4,0)

6. Dec 16, 2016

Staff: Mentor

With a complex z and c:$$\frac c z = \frac{cz^*}{zz^*}$$
Here * is the complex conjugation. Now the denominator is real and you can split the fraction into real and imaginary part.

7. Dec 17, 2016