Question

Find the modulus of $\frac{1+2i}{1-3i}$

Answer 1

Let $z=\frac{1+2i}{1-3i}$

Rationalising, we get

$=\frac{1+2i}{1-3i}\times \frac{1+3i}{1+3i}$

$=\frac{1+3i+2i+6i^2}{1-9i^2}$

$=\frac{1+5i-6}{1+9}$

$=\frac{-5+5i}{10}$

$=\frac{-1}{2}+\frac{i}{2}$

$\therefore z=\frac{-1}{2}+\frac{i}{2}$

Modulus of $z=\left|z\right|=\sqrt{{\left(\frac{-1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^2}=\sqrt{\frac{1}{4}+\frac{1}{4}}=\frac{1}{\sqrt{2}}$

$\therefore$ Modulus of $z=\frac{1}{\sqrt{2}}$