- #1
Alettix
- 177
- 11
Homework Statement
Consider the relation ## |\frac{z-i}{z*-i}| = \lambda ## where z = x + yi
a) For ##\lambda = 1## show that the locus is a line in the complex plane and find its equation.
b) What is the locus when ##\lambda = 0##?
c) Show that for all other positive ##\lambda## the locus may be written as ##zz* + bz* + b*z + c = 0## where ##c## is a real number. Find conditions on complex #b# and real #c# for which the expression describes a
i) Point
ii) Circle
iii) Line
Homework Equations
## \frac{m}{n} = \frac{mn*}{|n|^2}##
if ## m = k + ip ##, ##|m| = \sqrt{k^2+p^2}##
The Attempt at a Solution
I tried the following on the first part:
## 1 = |\frac{z-i}{z*-i}| = |\frac{x+i(y-1)}{x-i(y+1)}| = |\frac{(x+i(y-1))\cdot (x+i(y+1))}{x^2+(y+1)^2}| = |\frac{x^2-y^2-2xyi}{x^2+(y+1)^2}| ##
but from here I am stuck, because this does not resemble the equation of a line at all.
Thanks for all help and hints! :)