Question

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

• A. $\left[r\right]\ne 2$
• B. $0<r<3$
• C. $r=6$
• D. $3<r<2\sqrt{3}$

Answer 1

Answer: (A), (D)

$\left[r\right]\ne 2$
$3<r<2\sqrt{3}$

$CP=r,OC=2\sqrt{3},\angle COP=\frac{\pi }{3}$
$\Rightarrow CP=OCsin\frac{\pi }{3}=2\sqrt{3}\frac{\sqrt{3}}{2}=3$
Thus, when $r=3$, the circle touches the line.

Hence, for two distinct points of intersection, $3<r<2\sqrt{3}$.