Search any question & find its solution
Question:
Answered & Verified by Expert
If $Z_1=2+i$ and $Z_2=3-4 i$ and $\frac{\overline{Z_1}}{Z_1}=a+b i$, then the value of $-7 a+b$ is (where $i=\sqrt{-1}$ and $\mathrm{a}, \mathrm{b} \in \mathbb{R})$
Options:
Solution:
1108 Upvotes
Verified Answer
The correct answer is:
-1
$\begin{aligned} & \mathrm{Z}_1=2+\mathrm{i} \quad \mathrm{Z}_2=3-4 \mathrm{i} \\ & \overline{Z_1}=2-\mathrm{i} \quad \overline{Z_2}=3+4 \mathrm{i} \\ & \frac{\overline{Z_1}}{\overline{Z_2}}=\frac{2-\mathrm{i}}{3+4 \mathrm{i}} \\ & =\frac{(2-i)(3-4 i)}{(3+4 i)(3-4 i)} \\ & =\frac{6-8 i-3 i+4 i^2}{(3)^2-(4 i)^2} \\ & =\frac{6-1 \mathrm{li}-4}{9+16} \quad \ldots\left[\mathrm{i}^2=-1\right] \\ & \mathrm{a}+\mathrm{bi}=\frac{2-1 \mathrm{li}}{25} \\ & \therefore \quad a=\frac{2}{25}, b=\frac{-11}{25} \\ & \end{aligned}$
$\begin{aligned} & \text { Now }-7 \mathrm{a}+\mathrm{b} \\ & =-7\left(\frac{2}{25}\right)-\frac{11}{25} \\ & =\frac{-14-11}{25} \\ & =\frac{-25}{25}=-1\end{aligned}$
$\begin{aligned} & \text { Now }-7 \mathrm{a}+\mathrm{b} \\ & =-7\left(\frac{2}{25}\right)-\frac{11}{25} \\ & =\frac{-14-11}{25} \\ & =\frac{-25}{25}=-1\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.