Question

Let $z_1$ and $z_2$ be two complex numbers such that $z_1\neq z_2$ and $|z_1|\neq| z_2|$ . If $z_1$ has a positive real part and $z_2$ has negative imaginary part, then $\frac{z_1+z_2}{z_1+z_2}$ may be

• A. zero or purely imaginary
• B. real and positive
• C. real and negative
• D. none of these

Answer 1

Answer: (C)

Let $z_{1}=cos\,\theta+i\,sin\,\theta$ and $z_{2}=cos\,\phi+i\,sin\,\phi$
$\therefore \frac{z_{1}-z_{2}}{z_{1}-z_{2}}$

$=\frac{cos\,\theta+i\,sin\,\theta+cos\,\phi+i\,sin\,\phi}{cos\,\theta+i\,sin\,\theta-cos\,\phi-i\,sin\,\phi}$

$=\frac{\left(cos\,\theta+\,+cos\,\phi\right)+i\left(sin\,\theta+sin\,\phi\right)}{\left(cos\,\theta-cos\,\phi\right)+i\left(sin\,\theta-sin\,\phi\right)}$

$=-i\,cot \frac{\theta-\phi}{2}$

which is purely imaginary if $\theta \ne\phi$ and zero if

$\frac{\theta-\phi}{2}=\frac{\pi}{2}$ or $\theta=\pi+\phi$