The complex number having least positive argument and satisying `|z-5i| le 3`, is

Options

A. `12+16i`
B. `12/5 + (16i)/5`
C. `16/5+(12i)/5`
D. `-12/5+(16i)/5`

Answer

B

Explanation

The region represented by `|z-5i| le 3` is the region containing all points lying inside or on the circle having center at (0,5) and radius 3. Clearly, the complex number having the least positive argument in this region is the complex number represented by the point of contact of tangent drawn from the origin to the circle.

Clearly, `OP = sqrt((OC^(2)-CP^(2))=sqrt(25-9))=4`
So, the coordinate of P are `(4costheta,4sintheta)`.
From triangle OCP, we have `costheta=3/5`
So, the coordinates of P are `(12//5,16//15)`.
Hence, required complex number is `12/5+16/5i`.