Question: $z_1,z_2$ are two complex numbers with $z_2 \neq 0$ and $z_1 \neq z_2$ and satisfying: $$\left | \frac{z_1 + z_2}{z_1 - z_2}\right | = 1$$ Then $z_1\over z_2$ is:
A) Real and negative
B) Real and Positive
C) Purely imaginary
D) None of the above
I think the answer should be (D) as we are only given the condition that $z_2 \neq 0$. If both $z_1,z_2 \neq 0$ then the answer would have been (C). Am I right?
If $a=\dfrac{z_1}{z_2}=x+iy$
we have $|a-1|=|a+1|\iff (x-1)^2=(x+1)^2\iff x=0$
A more lenghty approach, with $z_1=a_1+b_1\cdot i$ and $z_2=a_2+b_2\cdot i$:
$$ \begin{align} \frac{|z_1+z_2|}{|z_1-z_2|} & = 1 \newline |z_1+z_2| & = |z_1-z_2| \newline \sqrt{(a_1+a_2)^2+(b_1+b_2)^2} & = \sqrt{(a_1-a_2)^2+(b_1-b_2)^2} \newline (a_1+a_2)^2+(b_1+b_2)^2 & = (a_1-a_2)^2+(b_1-b_2)^2 \newline a_1^2+2a_1a_2+a_2^2+b_1^2+2b_1b_2+b_2^2 & = a_1^2-2a_1a_2+a_2^2+b_1^2-2b_1b_2+b_2^2 \newline 2a_1a_2+2b_1b_2 & = -2a_1a_2-2b_1b_2 \newline 4a_1a_2+4b_1b_2 & = 0 \newline a_1a_2+b_1b_2 & = 0 \newline \end{align} $$ So, let's look at $\frac{z_1}{z_2}$: $$ \frac{z_1}{z_2} = \frac{a_1+b_1\cdot i}{a_2+b_2\cdot i} = \left(\frac{a_1a_2+b_1b_2}{a_2^2+b_2^2}\right)+\left(\frac{b_1a_2-a_1b_2}{a_2^2+b_2^2}\right)\cdot i \\ $$ Plugging in $a_1a_2+b_1b_2 = 0$ leads to: $$ \left(\frac{0}{a_2^2+b_2^2}\right)+\left(\frac{b_1a_2-a_1b_2}{a_2^2+b_2^2}\right)\cdot i = 0+\left(\frac{b_1a_2-a_1b_2}{a_2^2+b_2^2}\right)\cdot i $$ So, answer C) is correct.
