Find a and b if : 1 a+i b =3-2 i - Vidyadip

Question

Find a and b if :
$\frac{1}{a+i b}=3-2 i$

✓

Answer

$ \frac{1}{a+i b}=3-2 i$
$\therefore a+i b=\frac{1}{3-2 i}$
$\therefore a+i b=\frac{1}{3-2 i} \times \frac{3+2 i}{3+2 i}$
$\therefore a+i b=\frac{3+2 i}{3^2-2^2 i^2}$
$\therefore a+i b=\frac{3+2 i}{9-4(-1)} \quad \ldots\left[\because i^2=-1\right]$
$\therefore a+i b=\frac{3+2 i}{13}$
$\therefore a+i b=\frac{3}{13}+\frac{2}{13} i$
Equating real and imaginary parts, we get
$\therefore a =\frac{3}{13}$ and $b =\frac{2}{13}$

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