Question

In which quadrant of the complex plane, the point $\left( {\dfrac{{1 + 2i}}{{1 - i}}} \right)$ lies?
(A) Fourth
(B) First
(C) Second
(D) Third

Answer 1

Hint: In the given problem, we are required to find the quadrant in which the given expression involving complex numbers lies. So, we first simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator of the original complex number. Algebraic rules and properties also play a significant role in simplification of such expressions.

Complete answer:
In the question, we are given an expression involving complex numbers as $\left( {\dfrac{{1 + 2i}}{{1 - i}}} \right)$.
So, this expression first needs to be simplified using the knowledge of complex number sets and algebraic rules.
For simplifying the given expression involving complex numbers, we need to first multiply the numerator and denominator with the conjugate of the complex number present in the denominator so as to obtain a real number in the denominator.
So, we get,
$ \Rightarrow \left( {\dfrac{{1 + 2i}}{{1 - i}}} \right) \times \left( {\dfrac{{1 + i}}{{1 + i}}} \right)$
Using the algebraic identity $\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)$,
$ \Rightarrow \dfrac{{\left( {1 + 2i} \right)\left( {1 + i} \right)}}{{{{\left( 1 \right)}^2} - {{\left( i \right)}^2}}}$
Now, multiplying the brackets in numerator, we get,
$ \Rightarrow \dfrac{{1 + 2i + i + 2{i^2}}}{{\left( 1 \right) - \left( { - 1} \right)}}$
Simplifying the expression further, we get,
$ \Rightarrow \dfrac{{1 + 3i + 2\left( { - 1} \right)}}{2}$
$ \Rightarrow \dfrac{{ - 1 + 3i}}{2}$
Now, we notice that the real part of the complex number is negative and the imaginary part 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. So, a complex number with negative real part and positive imaginary part will lie in the second quadrant in the complex plane.
Hence, the option (C) is correct.

Note:
The given problem revolves around the application of properties of complex numbers in questions. The question tells us about the wide ranging significance of the complex number set and its properties. Algebraic rules and properties also play a significant role in simplification of such expressions and we also need to have a thorough knowledge of complex number sets and their applications in such questions. We should know the representation of x-axis and y-axis on the complex plane to solve the given problem.