Question

Find the modulus of $\frac{1-i}{\sqrt{5}-i\sqrt{3}}$.

Answer 1

Now,

$\frac{1-i}{\sqrt{5}-i\sqrt{3}}$

$=\frac{(1-i)(\sqrt{5}+i\sqrt{3})}{(\sqrt{5}{)}^2+(\sqrt{3}{)}^2}$

$=\frac{(\sqrt{5}+\sqrt{3})+i(\sqrt{3}-\sqrt{5})}{8}$.

Now the modulus of the given complex number is

$\frac{1}{8}\sqrt{(\sqrt{5}+\sqrt{3}{)}^2+(\sqrt{3}-\sqrt{5}{)}^2}$

$\frac{1}{8}\sqrt{2\left\{\right(\sqrt{5}{)}^2+\left(\sqrt{3}{)}^2\right\}}$

$=\frac{1}{8}\times \sqrt{16}$

$=\frac{1}{2}$