Question

If a complex number $z$ satisfies $|2z+10+10i|\le 5\sqrt{3}-5$, 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 $-\frac{5\pi }{6}$

$|2z+10+i10|\le 5\sqrt{3}-5$
$⟹|z+5+i5|\le \frac{5\sqrt{3}-5}{2}$
Locus of z is inner region of the above circle with center $(-5,5)\&radius\frac{5\sqrt{3}-5}{2}$
From the figure:
Least possible argument of z will be the slope of the tangent (line segment AC) to the circle from the origin in anti-clockwise sense.
In $△$CAB:
$\measuredangle CAB=\arcsin \frac{BC}{AB}=\arcsin \frac{(\sqrt{3}-1)}{2\sqrt{2}}=\frac{\Pi }{12}$
$\measuredangle Y^{\prime }OB=\frac{\Pi }{4}$
$\measuredangle XOC=\frac{\Pi }{2}+\measuredangle Y^{\prime }OB+\measuredangle CAB=\frac{\Pi }{2}+\frac{\Pi }{4}+\frac{\Pi }{12}=\frac{5\Pi }{6}$
Least possible arg(z) is $\frac{-5\Pi }{6}$
Ans: A