Question

The real part of the multiplicative inverse of $\frac{1+2i}{5+6i}$ is

• A. $\frac{17}{61}$
• B. $\frac{5}{17}$
• C. $\frac{17}{5}$
• D. $\frac{17}{8}$

Answer 1

The correct option is C $\frac{17}{5}$

Let the multiplicative inverse be $Z$, then
$Z\times \frac{1+2i}{5+6i}=1\Rightarrow Z=\frac{1}{\frac{1+2i}{5+6i}}\Rightarrow Z=\frac{5+6i}{1+2i}\times \frac{1-2i}{1-2i}\Rightarrow Z=\frac{5-10i+6i-12i^2}{1^2-(2i{)}^2}\Rightarrow Z=\frac{17-4i}{1+4}=\frac{17-4i}{5}\therefore Re\left(Z\right)=\frac{17}{5}$