Theory of Intelligence
11TH•Mathematics•Complex Numbers•Question 4

Solution

If $z_1=2+3i$ and $z_2=1-ai$, find the real value of $a$ such that $\operatorname{Im}(z_1z_2)=7$.

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Solution

Given:

z1=2+3i,z2=1−aiz_1=2+3i,\quad z_2=1-ai

Compute z1z2z_1z_2:

z1z2=(2+3i)(1−ai)=2−2ai+3i−3ai2=2−2ai+3i+3a=(2+3a)+i(3−2a)\begin{aligned} z_1z_2&=(2+3i)(1-ai) \\ &=2-2ai+3i-3ai^2 \\ &=2-2ai+3i+3a \\ &=(2+3a)+i(3-2a) \end{aligned}

So:

Im⁡(z1z2)=3−2a\operatorname{Im}(z_1z_2)=3-2a

Given Im⁡(z1z2)=7\operatorname{Im}(z_1z_2)=7:

3−2a=7⇒−2a=4⇒a=−23-2a=7\Rightarrow -2a=4\Rightarrow a=-2 a=−2\boxed{a=-2}

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