If $z_1,z_2,z_3$ are the vertices of an equilateral triangle with centroid at origin & length of  circumradius of triangle is 1 unit. Then which of the following may be the vertices of an equilateral triangle? 

For equilateral $\Delta$ 
$z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$ 
A) Which is satisfied by $-z_1-z_2 \& z_3$ also Similarly $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$
B) $z_1+1,z_2+1,z_3+1$ are also satisfying the relation 
 
Are also vertices of equilateral $\Delta$  
D) $\frac{z_1}{2},\frac{z_2}{2},\frac{z_3}{2}$ are also vertices of an equilateral triangle  

$v\left(x\right)=\pi {(x+1)}^2\times 1+\pi h\{{(x+1)}^2-x^2\}$ 
$=\pi {(x+1)}^2+\pi h(2x+1)$ ……….(i)
Also $27\pi =\pi x^{2}h$  
$h=\frac{27}{x^2}$ …………(ii)
From (i) and (ii)
$v\left(x\right)=\pi {(x+1)}^2+\frac{27\pi }{x^2}(2x+1)=\pi \{{(x+1)}^2+\frac{54}{x}+\frac{27}{x^2}\}$ 
$a=1, b=54, c=27$    
$\frac{a.b}{c}=2\Rightarrow$ option(C)