Question

Let one of the vertices of the equilateral triangle circumscribing the circle $|z-2 \sqrt{3} i|=1$ is $z_1=1+3 \sqrt{3} i$. If the other two vertices are represented by $z_2$ and $z_3$, then which of the following statement(s) is/are CORRECT?

• A. The area of triangle is $\frac{3 \sqrt{3}}{4}$.
• B. $\left|\operatorname{Re}\left( z _2 z _3\right)\right|+\left| I _{ m }\left( z _2 z _3\right)\right|=8 \text {. }$
• C. The radius of circle circumscribing the triangle is 2 .
• D. The perimeter of the escribed circle drawn opposite to vertex $z_1$ of triangle is $\frac{3 \pi}{2}$.

Answer 1

Answer: (B), (C)

By using rotation, we get $\frac{ z -2 \sqrt{3} i }{(1+3 \sqrt{3} i )-2 \sqrt{3} i }= e ^{ \pm \frac{ i 2 \pi}{3}}$
$\Rightarrow z =2 \sqrt{3} i +(1+ i \sqrt{3})\left(\frac{-1}{2} \pm \frac{ i \sqrt{3}}{2}\right) $
$\therefore z =-2+2 \sqrt{3} i , 1+ i \sqrt{3}$
Clearly radius of circle circumscribing the triangle is $2 \Rightarrow$ option $( C )$ is correct
Now $z _2 z _3=(-2+2 \sqrt{3} i )(1+\sqrt{3} i )=-2-6-2 \sqrt{3} i +2 \sqrt{3} i =-8+ i 0$

$\Rightarrow \left|\operatorname{Re}\left( z _2 z _3\right)\right|+\left| I _{ m }\left( z _2 z _3\right)\right|=8$
$\Rightarrow $ Option $( B )$ is correct.
Also area $(\triangle ABC )=\frac{\sqrt{3}}{4}(2 \sqrt{3})^2=3 \sqrt{3}$
$\Rightarrow $ Option (A) is incorrect.
Now $r_1=\frac{\Delta}{ s - a }=\frac{3 \sqrt{3}}{\frac{6 \sqrt{3}}{2}-2 \sqrt{3}}=\frac{3 \sqrt{3}}{\sqrt{3}}=3$
Hence the perimeter of the escribed circle $=6 \pi \Rightarrow$ Option (D) is incorrect