Discussion
Let \(z = \dfrac{a - \frac{i}{2}}{i - 2}\), where a is a real number and \(i = \sqrt{-1}\). If Im(z)=0 , then the value of a is equal to
Answer: A
The value of a is 1.
Multiply numerator and denominator by the conjugate of \(i - 2\), which is \(-i - 2\):
\[ \begin{aligned} \left(a - \tfrac{i}{2}\right)(-2 - i) &= -2a - ai + i + \tfrac{i^{2}}{2} \\ &= \left(-2a - \tfrac{1}{2}\right) + (1 - a)i \end{aligned} \]
The denominator becomes \((i - 2)(-i - 2) = 5\), a real number, so \(\text{Im}(z) = \frac{1 - a}{5}\).
Setting this to 0 gives a = 1. Check: with a = 1, z = \(\frac{1 - i/2}{i - 2} = -\frac{1}{2}\), which is real.
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