Finding the Center and Radius of a Circle with Complex Numbers and Loci

[Kajan thana]

Homework Statement

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

Homework Equations

args(x/y)=args(x)-args(y)
Circle theorem - inclined angle theorem

The Attempt at a Solution

I sketched the circle with major arc.
Radius= using Pythagorus I got the radius as 5 unit^2 .
H=O/sinθ . H=2.5/sin(π/6)

I am stuck on finding the centre point.

[/B]

 

[mfb]

Can you find one or two points on the circle?

 

[Ray Vickson]

Homework Statement

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

Homework Equations

args(x/y)=args(x)-args(y)
Circle theorem - inclined angle theorem

The Attempt at a Solution

I sketched the circle with major arc.
Radius= using Pythagorus I got the radius as 5 unit^2 .
H=O/sinθ . H=2.5/sin(π/6)

I am stuck on finding the centre point.
[/B]

Your radius is wrong.

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 $B(x,y)$?

 

[Kajan thana]

Your radius is wrong.

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 $B(x,y)$?

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

 

[Kajan thana]

Can you find one or two points on the circle?

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

 

[mfb]

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

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.

 

[Kajan thana]

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.

I finally got the answer right and the radius is 5 unit^2. Your way gave me the same answer as well.
Thank you so much.