The complex number $\dfrac{{\left( {1 + 2i} \right)}}{{\left( {1 - 2i} \right)}}$ lies in which quadrant of the complex plane.
A. First
B. Second
C. Third
D. Fourth

Hint: Here, in the given question, we need to find the quadrant in which the given expression involving a complex number lies. First, we will simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator of the original complex number. After this, we will apply algebraic rules to simplify the expression.

Complete step by step solution:
Given that, $\dfrac{{\left( {1 + 2i} \right)}}{{\left( {1 - 2i} \right)}}$
First, find the conjugate of $1 - 2i$.
Conjugate of complex numbers $z = a + ib$ is given by $\overline z = a - ib$.
Thus, the conjugate of $1 - 2i$ is $1 + 2i$.
Now, multiply the numerator and denominator of the given expression by the conjugate of the denominator.
$ \Rightarrow \dfrac{{1 + 2i}}{{1 - 2i}} \times \dfrac{{1 + 2i}}{{1 + 2i}}$
Now, use algebraic identity $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$
$ \Rightarrow \dfrac{{\left( {1 + 2i} \right)\left( {1 + 2i} \right)}}{{{{\left( 1 \right)}^2} - {{\left( {2i} \right)}^2}}}$
Now, multiply the brackets in numerator
$ \Rightarrow \dfrac{{1 + 2i + 2i + 2i \times 2i}}{{{{\left( 1 \right)}^2} - {{\left( {2i} \right)}^2}}}$
$ \Rightarrow \dfrac{{1 + 4i + 4{i^2}}}{{1 - 4{i^2}}}$
Substitute the value of ${i^2} = - 1$ in the above written expression
$ \Rightarrow \dfrac{{1 + 4i + 4\left( { - 1} \right)}}{{1 - 4\left( { - 1} \right)}}$
On simplifying the expression further, we get
$ \Rightarrow \dfrac{{1 + 4i - 4}}{{1 + 4}}$
$ \Rightarrow \dfrac{{ - 3 + 4i}}{5}$
Now, we can see that the real part of the complex number is negative and the imaginary part of the complex number is positive. We know that the $x - axis$ on the complex number plane represents the real number line and $y - axis$ represents the imaginary number line. Hence, the complex number $\dfrac{{\left( {1 + 2i} \right)}}{{\left( {1 - 2i} \right)}}$ lies in the second quadrant of the complex plane.

Option ‘B’ is correct

Note: To solve this type of questions, one must know the representation of $x - axis$ and $y - axis$ on the complex plane. The given problem revolves around the application of properties of complex numbers. Remember the following rules to determine in which quadrant the complex numbers belongs:
For any complex number $z = x + iy$,
- If $x > 0$ and $y > 0$, then the complex number will lie in the first quadrant
- If $x < 0$ and $y > 0$, then the complex number will lie in the second quadrant
- If $x < 0$ and $y < 0$, then the complex number will lie in the third quadrant
- If $x > 0$ and $y > 0$, then the complex number will lie in the fourth quadrant