Question

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

Answer 1

Here $-6-24i=-6+24i$

Now $(x-iy)(3+5i)=-6+24i$

$\Rightarrow 3x+5xi-3yi-5y{i}^2=-6+24i$

$\Rightarrow (3x+5y)+(5x-3y)i=-6+24i$

Comparing both sides, we have

$3x+5y=-6\dots \left(i\right)$
and $5x-3y=24\dots \left(ii\right)$

Multiplying (i) by 3 and (ii) by 5 and then adding

$9x+15\text{/}y=-18\underset{–}{25x-15\text{/}y=120}34x=102\Rightarrow x=3$

Putting x=3 in (i)

$3\times 3+5y=-6\Rightarrow 5y=-6-9$

$\Rightarrow y=\frac{-15}{5}=-3$