Question

Let $z$ be a complex number satisfying $|2z+10+10i|\le 5\sqrt{3}-5.$ If $\arg$ denotes the principal argument lying in $(-\pi ,\pi ],$ then the least value of $\arg \left(z\right)$ is

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

Answer 1

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

$|2z+10+10i|\le 5\sqrt{3}-5$
$\Rightarrow |z+(5+5i)|\le \frac{5\sqrt{3}-5}{2}$
This represents interior of a circle whose centre is $(-5,-5)$ and radius is $\frac{5}{2}(\sqrt{3}-1).$

Here, $AB=\text{radius}=\frac{5}{2}(\sqrt{3}-1)$
$OA=\sqrt{5^2+5^2}=5\sqrt{2}$
$\sin \theta =\frac{\frac{5}{2}(\sqrt{3}-1)}{5\sqrt{2}}=\frac{\sqrt{3}-1}{2\sqrt{2}}$
$\therefore \theta ={15}^{\circ }=\frac{\pi }{12}$
$\arg \left(z\right)$ is minimum when $z$ is at $B.$
$\angle XOA=-\pi +{\tan }^{-1}\left|\frac{-5}{-5}\right|=-\frac{3\pi }{4}$
$\angle XOB=\angle XOA+(-\theta )$
$=-\frac{3\pi }{4}-\frac{\pi }{12}=\frac{-5\pi }{6}$