Question

Convert the following complex numbers into polar form.
$\frac{5-i}{2-3i}$

• A. $\sqrt{2}{e}^{i\frac{\pi }{4}}$
• B. $\sqrt{2}{e}^{i\frac{\pi }{2}}$
• C. $e^{i\frac{\pi }{4}}$
• D. $e^{i\frac{\pi }{2}}$

Answer 1

The correct option is A $\sqrt{2}{e}^{i\frac{\pi }{4}}$

We have, $\frac{5-i}{2-3i}$

$=\frac{5-i}{2-3i}\times \frac{2+3i}{2+3i}$

$\Rightarrow \frac{(10+3)+13i}{13}=1+i$

$\Rightarrow \sqrt{2}(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}})$

$\Rightarrow \sqrt{2}(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})$

$=\sqrt{2}{e}^{\frac{i\pi }{4}}$.