Question

The real values of $\mathrm{x}$ and $y$ for which the equation is $(\mathrm{x}+\mathrm{iy})(2-3\mathrm{i})=4+\mathrm{i}$ is satisfied, are

• A. $x=\frac{5}{13},y=\frac{8}{13}$
• B. $x=\frac{8}{13},y=\frac{5}{13}$
• C. $x=\frac{5}{13},y=\frac{14}{13}$
• D. None of these

Answer 1

The correct option is C

$x=\frac{5}{13},y=\frac{14}{13}$

Compute the required values:

Given, $(\mathrm{x}+\mathrm{iy})(2-3\mathrm{i})=4+\mathrm{i}$

$$\begin{gathered} \Rightarrow 2\mathrm{x}+\mathrm{i}2\mathrm{y}-\mathrm{i}3\mathrm{x}+3\mathrm{y}=4+\mathrm{i} \\ \Rightarrow 2\mathrm{x}+3\mathrm{y}+\mathrm{i}(2\mathrm{y}-3\mathrm{x})=4+\mathrm{i} \end{gathered}$$

Comparing real and imaginary parts,

$2x+3y=4$……………..$\left(1\right)$

$-3x+2y=1$……………..$\left(2\right)$

Add $\left[3\times \left(1\right)\right]$ and $\left[2\times \left(2\right)\right]$,

$6x+9y-6x+4y=12+2$

$\Rightarrow$ $13y=14$

$\Rightarrow$ $y=\frac{14}{13}$

Substitute the value of $y$ in equation$\left(1\right)$,

$2x+3\left(\frac{14}{13}\right)=4$

$\Rightarrow$ $2x=\frac{52-42}{13}$

$\Rightarrow$ $x=\frac{5}{13}$

Hence real value of $x$ and $y$ is $\frac{5}{13}$ and $\frac{14}{13}$.

Hence, option (c) is the correct option.