Question

If a line makes an angle $\theta (\ne \pi /2)$ with the positive direction of the $x-axis$, then its complex slope equals

• A. $e^{2i\theta }$
• B. $e^{-2i\theta }$
• C. $i{e}^{2i\theta }$
• D. None of these

Answer 1

The correct option is A $e^{2i\theta }$

Let the line formed be by joining $z_1$ and origin. Let $arg\left(z_1\right)=\theta$
Complex slope of the line joining $z_1$ and origin is: $\frac{z_1-0}{\overline{z_1}-0}=\frac{z_1}{\overline{z_1}}=\frac{|z_1|e^{i\theta }}{|z_1|e^{-i\theta }}=e^{2i\theta }$
Hence, (A) is correct.