Question

Let $z_1$ & $z_2$ be complex numbers such that $z_1\ne z_2$ & $|z_1|=|z_2|$. If $z_1$ has positive real part and $z_2$ has negative imaginary part, then $\frac{z_1+z_2}{z_1-z_2}$ may be

• A. zero
• B. real and positive
• C. real and negative
• D. purely imaginary
  

Answer 1

Answer: (A), (D)

The correct options are
A zero
D purely imaginary

We have $|Z_1|=|Z_2|$.
Consider vectors along the position vector of $Z_1$ and $Z_2$ in the Argand Plane and with magnitude equal to $|Z_1|$ and $|Z_2|$. Consider the rhombus formed as shown in the figure.
the vector along $Z_1+Z_2$ will be in the direction of one of the diagonals, while $Z_1-Z_2$ will be along the direction of the other diagonal.
The diagonals of a rhombus are perpendicular to each other. Hence, $\frac{Z_1+Z_2}{Z_1-Z_2}$ will have argument of $\pm \frac{\pi }{2}$. Hence, $\frac{Z_1+Z_2}{Z_1-Z_2}$ will be imaginary.
When $Z_1=-Z_2$ rhombus wont be formed but the expression would be zero.