Question

Let S be the set of all complex numbers `z` satisfying `| -2+i|ge sqrt(5)`.if the complex number `z_(0)` is such that `(1)/(|z_0-1|)` is the maximum of the set `{(1)/(|z-1|): ζinS}`, then the principal argument of `(4-z_0-overline(z)_0)/(z-overline(z)_0+2i)` is
A. `(pi)/(4)`
B. `(3pi)/(4)`
C. `-(pi)/(2)`
D. `(pi)/(2)`

Answer 1

Correct Answer - C
The complex number ` z` satisfying `| z-2 +i|ge sqrt(5)`, which represents the region outside the circle (including the circumference )having centre `(2,-1)` and radius `sqrt(5)` units.

Now, for ` z_(0) in S(1)/(| z_(0)-1|)` is maximum.
When `| z_0-1|` is minimum. And for this it is required that `z_0` is collinear with the points `(2,-1)` and `(1,0)` and lies on the circumference of the circle `|ᶼ-2+i|=sqrt(5)`.
So let `z_0=x+iy`, and the figure `0lt x lt 1 and y gt 0`.
So, `(4-z_(0)-overline(z)_(0))/(z_0-overline(z)_(0)+2i)=(4-x-iy-x+iy)/(x+iy-x+iy+2i)=(2(2-x))/(2i(y+1))=-i((2-x)/(y+1))`
`because (2-x)/(y+1)` is a positive real number, so
`(4-z_0-overline(z)_0)/(z_0-overline(z)_0+2i)` is purely negative imaginary number.
`rArr arg ((4-z_0-overset(-)z_(0))/(z_0-overset(-)z_(0)+2i))=-(pi)/(2)` .