If `|z-1|+|z+3|<=8`, then the range of values of `|z-4|` is

Options

A. (0,8)
B. [0,9]
C. [1,9]
D. [5,9]

Answer

C

Explanation

We have,
`|z-1|+|z+3| le8`
`rArr |z-(1+0i)|+|z-(-3+0i)| le 8` …………….(i) ltbr. Thus, if P,S and `S^(')` are three points in the Argand plane representing complex numbers, z,1+0i, and `-3+0i`, then from (i) we obtain.
`PS+PS^(') le 8`
`rArr` P lies inside and on the ellipse whose two foci are at S(1,0) and `S^(')(-3,0)` and major axis =8.

Clearly, PQ `=|z-4|`
Clearly,PQ is minimum or maximum according as P coincides with A(3,0) and `A^(')(-5,0)` respectively.
Thus, PQ=`|z-4|` varies between AQ=1 and `A^(')Q=9`.
Hence, `|z-| in |1,9|`.