Question

Find the real numbers $x$ and $y$ , if $(x-iy)(3+5i)$ is the conjugate of $-6-24i$.

Answer 1

Let $z=(x-iy)(3+5i)$
$z=3x+5xi-3yi-5{yi}^2=3x+5xi-3yi+5y=(3x+5y)+i(5x-3y)$
$\therefore \overline{z}=(3x+5y)-i(5x-3y)$
It is given that, $\overline{z}=-6-24i$
$\therefore (3x+5y)-i(5x-3y)=-6-24i$
Equating real and imaginary parts, we obtain
$3x+5y=-\mathrm{6.........}\left(i\right)$
$5x-3y=\mathrm{24..........}\left(ii\right)$
Multiplying equation (i) by $3$ and equation (ii) by $5$ and then adding them, we obtain
$9x+15y=-18$
$25x-15y=120$
$34x=102$
$\therefore x=\frac{102}{34}=3$
Putting the value of $x$ in equation (i) , we obtain
$3\left(3\right)+5y=-6$
$\Rightarrow 5y=-6-9=-15$
$\Rightarrow y=-3$
Thus, the values of $x$ and $y$ are $3$ and $-3$ respectively.