Question

If the imaginary part of \[\dfrac{{2 + i}}{{ai - 1}}\] is zero, where \[a\] is a real number, then the value of \[a\] is equal to
(\[A\]) \[1/3\]
(\[B\]) \[2\]
(\[C\]) \[ - 1/2\]
(\[D\]) \[ - 2\]

Answer 1

Hint: The given question is based on the topic of complex numbers. A complex number is the sum of real and imaginary parts. In this question, the value of the imaginary part is given as zero, and you need to find the value of the real number \[a\]. To find this value write the real part and imaginary part separately. And then by applying the imaginary part is zero. We can easily find out the value of \[a\].

Complete step by step answer:
We are given that \[\dfrac{{2 + i}}{{ai - 1}}\],
We know that in a complex number an imaginary part is followed by a real number. While here the real number is followed by an imaginary number in the denominator so we can simply rearrange denominator,
\[ = \dfrac{{2 + i}}{{ - 1 + ai}}\]
The given number is in fraction to find the value we can rationalize. In rationalization the conjugate of the denominator is multiplied by both the numerator and denominator.
\[ = \dfrac{{2 + i}}{{ - 1 + ai}} \times \dfrac{{ - 1 - ai}}{{ - 1 - ai}}\]
\[ = \dfrac{{(2 + i)( - 1 - ai)}}{{( - 1 + ai)( - 1 - ai)}}\]
Here denominator is in the form \[(a + b)(a - b)\] which is equal to \[{a^2} - {b^2}\]. Here let \[ - 1\] be \[a\] and \[ai\] be \[b\].
\[ = \dfrac{{(2 + i)( - 1 - ai)}}{{{{( - 1)}^2} - {{(ai)}^2}}}\]
Multiplying the values in the brackets in numerator and squaring the values in denominator we will get,
\[ = \dfrac{{ - 2 - 2ai - i - a{i^2}}}{{1 - {a^2}{i^2}}}\]
In complex number, \[{i^2} = - 1\], by substituting this,
 \[ = \dfrac{{ - 2 - 2ai - i - a( - 1)}}{{1 - {a^2}( - 1)}}\]
\[ = \dfrac{{ - 2 - 2ai - i + a}}{{1 + {a^2}}}\]
Now the denominator only has real numbers and in the numerator we have both real and imaginary parts so let's write real and imaginary parts separately.
\[ = \dfrac{{( - 2 + a) + ( - 2ai - i)}}{{1 + {a^2}}}\]
Taking \[ - i\] common for imaginary parts,
\[ = \dfrac{{( - 2 + a) - i(2a - 1)}}{{1 + {a^2}}}\]
Now we have imaginary part \[ - i(2a - 1)\] and it is given that the value of imaginary part is \[0\], that is, \[2a - 1 = 0\]
Now solving this \[2a - 1 = 0\], we will get the value of \[a\].
\[2a - 1 = 0\]
\[2a = - 1\]
\[a = - \dfrac{1}{2}\]
Therefore, the value of \[a = - \dfrac{1}{2}\].

Note:
We know that rationalization is the method used to multiply the fraction number with the conjugate of its denominator. Remember that according to complex numbers the conjugate is applicable only for the imaginary part and not for the real part. Here in our question the complex part is \[ai\] so changing its sign we will get the conjugate and the conjugate of \[ai - 1\] is, \[ - ai - 1\]. To avoid this confusion, in the first step itself we rewrite the denominator \[ai - 1\] into \[ - 1 + ai\], because generally we change the sign of the second term to take its conjugate.