Question

If the complex number $\dfrac{{8 - i}}{{3 - 2i}}$ is rewritten in the form of (a + ib), where a is the real part and b is the imaginary part, then what is the value of a? $({\text{Use i = }}\sqrt { - 1} )$
$
  (a){\text{ 2}} \\
  (b){\text{ }}\dfrac{8}{3} \\
  (c){\text{ 3}} \\
  (d){\text{ }}\dfrac{{11}}{3} \\
 $

Answer 1

Hint – In this question rationalize the given complex number by multiplying the numerator and the denominator part by conjugate of the original denominator that is $3 + 2i$. Then after simplification compare with the standard form of (a+ib), to get the value of a.

Complete step-by-step answer:
Given complex number is
$\dfrac{{8 - i}}{{3 - 2i}}$
Now first convert this in the form of (a + ib).
So, first rationalize the complex number, (i.e. multiply and divide by (3 + 2i) in the given complex number) we have,
$ \Rightarrow \dfrac{{8 - i}}{{3 - 2i}} \times \dfrac{{3 + 2i}}{{3 + 2i}}$
Now multiply the numerator and in denominator apply the rule $\left[ {\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}} \right]$
$ \Rightarrow \dfrac{{24 - 3i + 16i - 2{i^2}}}{{9 - 4{i^2}}}$
Now as we know in complex the value of $\left[ {{i^2} = - 1} \right]$ so, use this property in above equation we have,
$ \Rightarrow \dfrac{{24 - 3i + 16i - 2\left( { - 1} \right)}}{{9 - 4\left( { - 1} \right)}}$
Now simplify the above equation we have,
$ \Rightarrow \dfrac{{24 + 13i + 2}}{{9 + 4}} = \dfrac{{26 + 13i}}{{13}} = 2 + i$
So this is the required form.
$ \Rightarrow 2 + i = a + ib$
So on comparing we have,
$a = 2,b = 1$
Therefore, a = 2
So, this is the required answer.
Hence option (A) is correct.

Note – The real and imaginary part of a complex number is of great significance as it helps in plotting a complex number in the argand plane. The real part is plotted on the real axis whereas the imaginary part is plotted upon the complex axis. That’s why we express a complex number in terms of real and imaginary parts separately.