Complex Number Help (1 Viewer)

[Green Yoda]

Both Parts Thanks

 

[pikachu975]

part i:

From the first equation we know that the modulus of z-1 is half that of z, so when you divide (z-1)/z you divide moduli, giving 1/2.

For the angle, the question tells you that the angle between them is pi/3 and if you draw the diagram and minus arguments you can see that. Also when you divide (z-1)/z you minus arguments, and they tell you that it's pi/3 in the question, therefore cis(pi/3)

So then you get the answer of 1/2 (cispi/3)

part ii:

Not sure

 

[jathu123]

For part ii)

Turn the mod arg form into x+iy and simplify to get
4z-4=z(1+root3i). Rearrange to get
3z-z√3 i =4.
z(3-√3 i)=4. Now make z the subject
z=4/(3-√3 i) realising this, we get
Z= (4(3+ √3i))/(9+3). Simplify, then
Z=(3+root 3i)/3

Typing on phone so soz if it's messy

 

[braintic]

Do you know your locus work?
The first equation gives a full circle, the second gives a circular arc.
The answer to (ii) is the intersection of the circle with the arc.