Complex NumbersHard
Question
Let S = {z : z = x + iy, y ≥ 0, |z - z0| ≤ 1 }
where |z0| = |z0 - ω| = |z0 - ω2|, ωand ω2 are non real cube roots of unity, then -
where |z0| = |z0 - ω| = |z0 - ω2|, ωand ω2 are non real cube roots of unity, then -
Options
A.z0 = - 1
B.z0 = - 
C.if z ∈ S, then least value of |z| is 1
D.|amp(ω - z0)| = 
Solution
∵ z0 is circumcentre of triangle joining (0,0),
ω ≡
and ω2 ≡ 
⇒ z0 ≡ (-1, 0) {∵ | - 1 - 0 | = |- 1 ω| = |- 1 ω2|}
∵ z lies in |z + 1| ≤ 1
⇒ minimum value of |z| is 0
amp(ω - z0)= angle made by line joining z0 to ω with real axis
=
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