Question:
If $x+i y=\frac{3+5 i}{7-6 i}$, then $y=$
(a) 9/85
(b) −9/85
(c) 53/85
(d) none of these
Solution:
(c) $\frac{53}{85}$
$x+i y=\frac{3+5 i}{7-6 i}$
$\Rightarrow x+i y=\frac{3+5 i}{7-6 i} \times \frac{7+6 i}{7+6 i}$
$\Rightarrow x+i y=\frac{21+53 i+30 i^{2}}{49-36 i^{2}}$
$\Rightarrow x+i y=\frac{21-30+53 i}{49+36}$
$\Rightarrow x+i y=\frac{-9}{85}+i \frac{53}{85}$
On comparing both the sides :
$y=\frac{53}{85}$