Question

Find the modulus and amplitude of $\frac{2+6\sqrt{3}i}{5+\sqrt{3}i}$

• A. $|z|=4;amp\left(z\right)=\frac{5\pi }{3}$
• B. $|z|=2;amp\left(z\right)=\frac{\pi }{3}$
• C. $|z|=4;amp\left(z\right)=\frac{\pi }{3}$
• D. $|z|=2;amp\left(z\right)=\frac{5\pi }{3}$

Answer 1

The correct option is B $|z|=2;amp\left(z\right)=\frac{\pi }{3}$

Given $\frac{2+6\sqrt{3}i}{5+\sqrt{3}i}$

Multiplying and dividing $5-i\sqrt{3}$, we get

$2\frac{(1+3\sqrt{3}i)(5-i\sqrt{3})}{25+3}$
$=\frac{5+9+14\sqrt{3}i}{14}$
$=\frac{14+14\sqrt{3}i}{14}$

$=1+\sqrt{3}i$

$=2\left(\frac{1+\sqrt{3}i}{2}\right)$

$=2(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3})$

$=2e^{i\frac{\pi }{3}}$