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Question: If the complex number \(\dfrac{{8 - i}}{{3 - 2i}}\) is rewritten in the form of (a + ib), where a is...

If the complex number 8−i3−2i\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? (Use i = −1)({\text{Use i = }}\sqrt { - 1} )
(a) 2 (b) 83 (c) 3 (d) 113  (a){\text{ 2}} \\\ (b){\text{ }}\dfrac{8}{3} \\\ (c){\text{ 3}} \\\ (d){\text{ }}\dfrac{{11}}{3} \\\

Explanation

Solution

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+2i3 + 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
8−i3−2i\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,
⇒8−i3−2i×3+2i3+2i\Rightarrow \dfrac{{8 - i}}{{3 - 2i}} \times \dfrac{{3 + 2i}}{{3 + 2i}}
Now multiply the numerator and in denominator apply the rule [(a−b)(a+b)=a2−b2]\left[ {\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}} \right]
⇒24−3i+16i−2i29−4i2\Rightarrow \dfrac{{24 - 3i + 16i - 2{i^2}}}{{9 - 4{i^2}}}
Now as we know in complex the value of [i2=−1]\left[ {{i^2} = - 1} \right] so, use this property in above equation we have,
⇒24−3i+16i−2(−1)9−4(−1)\Rightarrow \dfrac{{24 - 3i + 16i - 2\left( { - 1} \right)}}{{9 - 4\left( { - 1} \right)}}
Now simplify the above equation we have,
⇒24+13i+29+4=26+13i13=2+i\Rightarrow \dfrac{{24 + 13i + 2}}{{9 + 4}} = \dfrac{{26 + 13i}}{{13}} = 2 + i
So this is the required form.
⇒2+i=a+ib\Rightarrow 2 + i = a + ib
So on comparing we have,
a=2,b=1a = 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.