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)?

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.