If z is a complex number such that `|z|geq2` , then the minimum value of `|z+1/2|` (1) is equal to `5/2` (2) lies in the interval (1, 2) (3) is strictly greater than `5/2` (4) is strictly greater than `3/2` but less than `5/2`

Options

A. is strictly greater than `5/2`
B. is strictly greater than `3/2` but less than `5/2`
C. is equal to `5/2`
D. lies in the interval (1,2)

Answer

D

Explanation

We observe that `S=|z:|z| ge 2)` is the set of all points lying on or outside the circle centred at the origin and radius 2 and `|z+1/2|=|z-(-1/2+0i)|` is the distance of any point in S from the point `(-1//2,0)`. It is evident from Fig. that AP is minimum when P coincides with B and in that case `AP=AB=3//2` which lies in the interval (1,2).