Question

If $x+iy=\frac{3+5i}{7-6i},$ then y =
(a) 9/85
(b) −9/85
(c) 53/85
(d) none of these

Answer 1

(c) $\frac{53}{85}$
$$\begin{gathered} x+iy=\frac{3+5i}{7-6i} \\ \Rightarrow x+iy=\frac{3+5i}{7-6i}\times \frac{7+6i}{7+6i} \\ \Rightarrow x+iy=\frac{21+53i+30i^2}{49-36i^2} \\ \Rightarrow x+iy=\frac{21-30+53i}{49+36} \\ \Rightarrow x+iy=\frac{-9}{85}+i\frac{53}{85} \\ \text{O}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{the}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{s}: \\ y=\frac{53}{85} \end{gathered}$$