No they aren't necessarily purely real or purely imaginary. z1=cai-cb, z2=a+bi. That gives z1/z2=ci
Two complex numbers z1 and z2 are taken such that |z1+z2|=|z1-z2|, and z2 not equal to zero.
Prove that z1/z2 is purely imaginary (has no real parts).
So I said there are 4 possibilities (ie z1=a or bi, and z2=bi or a), and worked out whether the modulus equality holds for each (it does). The corresponding theta for z1 and z2 is then either 0 or pi/2. Then:
z1/z2 = a/bi=-(a/b)*i
or
z1/z2=(b/a)*i
Is this about right? For instance I'm really not certain about the angles. If z1=a (ie real only), then its theta is 0 right? And if z2=bi (imaginary only), it's theta is (pi/2)?
Many thanks for your help in advance.
CP
I let z1=a+bi, and z2=c+di. Then the modulus equality becomes: (a+c)^2 + (b+d)^2 = (a-c)^2 + (b-d)^2. I worked this out as a first step. Then take z1/z2, and work this out. You should find that it is possible to substitute what you got in step one into this division, and the outcome should be purely imaginary.
Best
CP