Question

Find the real and imaginary parts of the complex number $\frac{a+ib}{a-ib}$

Answer 1

$=\frac{a+ib}{a-ib}\cdot \frac{a+ib}{a+ib}$, [rationalize the denominator]

$=\frac{(a+ib{)}^2}{a^2-i^2{b}^2}=\frac{a^2+i^2{b}^2+2iab}{a^2+b^2}$

$=\frac{a^2-b^2+\left(2ab\right)i}{a^2+b^2}$

$=\frac{a^2-b^2}{a^2+b^2}+\frac{2ab}{a^2+b^2}i$

Hence real part is $\frac{a^2-b^2}{a^2+b^2}$ and imaginary part is $\frac{2ab}{a^2+b^2}$