Question Number 51571 by Tawa1 last updated on 28/Dec/18
$$\mathrm{If}\:\:\:\alpha\:−\:\mathrm{j}\beta\:\:=\:\:\frac{\mathrm{1}}{\mathrm{a}\:−\:\mathrm{jb}}\:\:,\:\:\:\:\:\:\mathrm{where}\:\:\:\alpha,\:\beta,\:\mathrm{a},\:\mathrm{b}\:\:\mathrm{are}\:\mathrm{real},\:\mathrm{express}\:\:\mathrm{b}\:\:\mathrm{in}\:\mathrm{terms} \\ $$$$\mathrm{of}\:\:\alpha,\:\beta\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Answer}:\:\:\:\:\:\:\:\:\:\:\frac{−\:\beta}{\alpha^{\mathrm{2}} \:+\:\beta^{\mathrm{2}} \:−\:\mathrm{2}\alpha\:+\:\mathrm{1}} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 28/Dec/18
$${a}−{ib}=\frac{\mathrm{1}}{\alpha−{i}\beta} \\ $$$${a}−{ib}=\frac{\alpha+{i}\beta}{\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} }=\left(\frac{\alpha}{\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} }\right)+{i}\left(\frac{\beta}{\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} }\right) \\ $$$${b}=−\left(\frac{\beta}{\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} }\right) \\ $$$${i}\:{think}\:{so}... \\ $$
Commented by Tawa1 last updated on 28/Dec/18
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$