Question

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

• A. $-\frac{5\pi }{6}$
• B. $0$
• C. $-\frac{3\pi }{4}$
• D. $-\frac{2\pi }{3}$

Answer 1

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

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

It implies that $z$ lies on or inside the circle of radius $\frac{5(\sqrt{3}-1)}{2}$ and centre $(-5,-5).$


Point $B$ has least principle argument, where line $OB$ is a tangent to the circle.
Now,
$BC=\frac{5(\sqrt{3}-1)}{2},OC=5\sqrt{2},$

Let $\angle COB=\theta ,$ then
$\sin \theta =\frac{BC}{OC}\Rightarrow \sin \theta =\frac{\frac{5(\sqrt{3}-1)}{2}}{5\sqrt{2}}$
$\Rightarrow \theta =\frac{\pi }{12}$

$\therefore \arg (z{)}_{min}=-(\frac{\pi }{2}+\frac{\pi }{4}+\frac{\pi }{12})=-\frac{5\pi }{6}$