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4-1.Complex numbers
easy
If ${z_1} = 1 + 2i$ and ${z_2} = 3 + 5i$, and then $\operatorname{Re} \left( {\frac{{{{\bar z}_2}{z_1}}}{{{z_2}}}} \right)$ is equal to
A
$\frac{{ - 31}}{{17}}$
B
$\frac{{17}}{{22}}$
C
$\frac{{ - 17}}{{31}}$
D
$\frac{{22}}{{17}}$
Solution
(d) Given ${z_1} = 1 + 2i$, ${z_2} = 3 + 5i$
$\frac{{{{\bar z}_2}{z_1}}}{{{z_2}}} = \frac{{(3 – 5i)\,(1 + 2i)}}{{(3 + 5i)}} = \frac{{13 + i}}{{3 + 5i}}$
= $\frac{{13 + i}}{{3 + 5i}} \times \frac{{3 – 5i}}{{3 – 5i}} = \frac{{44 – 62i}}{{34}}$
Then $\operatorname{Re} \left( {\frac{{{{\bar z}_2}{z_1}}}{{{z_2}}}} \right) = \frac{{22}}{{17}}$.
Standard 11
Mathematics
