If `z_(1)` and `z_(2)` are the complex roots of the equation `(x-3)^(3) + 1=0`, then `z_(1) +z_(2)` equal to

Options

A. `0 le d lt 15/2`
B. `0 lt d le 15/2`
C. `0 le d le 17/2`
D. `0 lt d lt 17/2`

Answer

C

Explanation

`P(z_(1))` and `Q(z_(2))` are points on the circle `|z-3| le 4` and the ellipse `|z-1|=|z+1|=3` respectively. The center of the circle is at `(3,0)` and radius 4. The center of the ellipse is at the origin and foci at `S(1,0)` and `S^(')(-1,0)`. The major axis `AA^(')` is of length 3. Therefore, OA= `OA^(')=3/2`.
Clearly, d=PQ. The minimum value of d is 0 when P and Q coincide with L or M.

The maximum value of PQ is
`A^(')N=OA^(')+ON=3/2+7=17/2`
Hence, `0 le d le 17/2`.