Question

The complex number $\left(\frac{1+2i}{1-2i}\right)$ lies in which quadrant of the complex plane.

• A. First
• B. Second
• C. Third
• D. Fourth

Answer 1

The correct option is B

Second

Explanation for correct answer:

Finding the quadrant in which the point lies by simplifying the expression given for $z$:

Let $z=\left(\frac{1+2i}{1-2i}\right)$

Rationalize z, by multiplying its numerator and denominator by $\left(1+2i\right)$.

$$\begin{gathered} \Rightarrow z=\left(\frac{1+2i}{1-2i}\right)\times \frac{\left(1+2i\right)}{\left(1+2i\right)} \\ \Rightarrow z=\frac{1+4i^2+4i}{1-4i^2} \\ \Rightarrow z=\frac{4i-3}{5}(i^2=-1) \\ \Rightarrow z=\frac{-3}{5}+\frac{4i}{5} \end{gathered}$$

Here, the real part of the complex number is negative while imaginary part of the complex number is positive.

Thus, the point lies in Quadrant Second.

Hence, the correct answer is Option (B).