Q116 / 180 · COMEDK 2012 · Mathematics

The multiplicative inverse of $\frac{3+4 i}{4-5 i}$ is

  • A
    $\left(\frac{-8}{25}, \frac{31}{25}\right)$
  • Correct answer
    B
    $\left(\frac{-8}{25}, \frac{-31}{25}\right)$
  • C
    $\left(\frac{8}{25}, \frac{-31}{25}\right)$
  • D
    $\left(\frac{-8}{25}, \frac{31}{5}\right)$

Solution

$\frac{3+4 i}{4-5 i}$ Now, multiplicative inverse is $$ \begin{aligned} & \frac{4-5 i}{3+4 i} \times \frac{3-4 i}{3-4 i} \\ =& \frac{(4-5 i)(3-4 i)}{9+16} \\ =& \frac{12-16 i-15 i+20 i^{2}}{25}=\frac{-8-31 i}{25}=\left(-\frac{8}{25},-\frac{31}{25}\right) \end{aligned} $$

Hence, the correct answer is option: 'B'.

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