Question

If a complex number $z$ satisfies $|2z+10+10i|<5\sqrt{3}-5$, then the least principal argument of $z$ is

• A. $-\frac{5\pi }{6}$
• B. $-\frac{11\pi }{12}$
• C. $-\frac{3\pi }{4}$
• D. $-\frac{2\pi }{3}$

Answer 1

The correct option is A $-\frac{5\pi }{6}$

$|2z+10+10i|\ge 5\sqrt{3}-5$
$\Rightarrow |z+5+5i|\le \frac{5(\sqrt{3}-1)}{2}$


Point $B$ has least principal argument. Now,
$AB=\frac{5(\sqrt{3}-1)}{2}$
$OA=5\sqrt{2}$
$\angle AOB=\frac{\pi }{12}$
$\therefore \arg \left(z\right)=-\frac{5\pi }{6}$