Question

Let z1, z2, z3 be three vertices of an equilateral triangle circumscribing the circle `|z| = 1/2`. If `z1 = 1/2+ ((sqrt3)iota)/2` and z1, z2, z3 are in the anticlockwise sense then z2 is

Answer 1

`abs(z)=1/2`,it represents circle with circle center `(0.0)` r=1/2
by drawing figure we get, since, z_2 is anticlockwise z_1=-1/2
`z_1=1/2+sqrt(3)i/2`
and Argument (z)=`sqrt(3)`
`tan theta=sqrt(3)` `theta=60^@` `Abs(z)=sqrt((1/2)^2+(sqrt(3)/2)^2)=1` similarly `abs(z_1)=abs(z_2)=abs(z_3)`
So,`abs(z)`=1 and `z` lies on the real axis of the positive direction `:.` Ans=D