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Question:

Let S be the set of all complex numbers z satisfying |z - (2 - i)| ≥ 5. If the complex number z0 is such that |z0 - 1| is the maximum of the set {|z - 1| : z ∈ S}, then the principal argument of (4 - z0 - z0*) / (z0 - z0* + 2i) is

π/4

π/2

3π/4

-π/2

Solution:

Correct option is C.
|z - (2 - i)| ≥ 5
For |z0 - 1| to be minimum, z0 = x0 + iy is at point P as shown in figure
arg(4 - (z0 + z0*)/(z0 - z0* + 2i)) = arg(4 - 2x0/(2iy0 + 2i)) = arg(-i(2 - x0)/(y0 + 1)) = arg(-iλ) = 3π/4.