Question

Given that the two curves $\arg \left(z\right)=\frac{\pi }{6}$ and $|z-2\sqrt{3}i|=r$ intersect in two distinct points, then
([r] represents integaral part of $r$)

• A. $\left[r\right]=3$
• B. $2<r<3$
• C. $\left[r\right]=2$
• D. $3<r<2\sqrt{3}$

Answer 1

Answer: (A), (D)

$3<r<2\sqrt{3}$



$CP=r,OC=2\sqrt{3},\angle COP=\frac{\pi }{3}$
$\Rightarrow CP=OC\sin \frac{\pi }{3}=2\sqrt{3}\frac{\sqrt{3}}{2}=3$
Thus, when $r=3$, the circle touches the line. Hence, for two distinct points of intersection $3<r<2\sqrt{3}$.