Let `C_1 and C_2` are concentric circles of radius ` 1and 8/3` respectively having centre at `(3,0)` on the argand plane. If the complex number `z` satisfies the inequality `log_(1/3)((|z-3|^2+2)/(11|z-3|-2))>1,` then (a) z lies outside `C_(1)` but inside `C_(2)` (b) z line inside of both ` C_(1)` and `C_(2)` (c) z line outside both `C_(1)` and `C_(2)` (d) none of these

Options

A. z lies outside `C_(1)` but inside `C_(2)`
B. z line inside of both ` C_(1)` and `C_(2)`
C. z line outside both `C_(1)` and `C_(2)`
D. none of these

Answer

A

Explanation

`log_(1//3)((|z-3|^(2)+2)/(11|z-3|-2))gt1,11|z-3|-2|gt0`
`rArr (|z-3|^(2)+2)/(11|z-3|-2)lt(1)/(3)`
`rArr (3t-8)(t-1)lt0 " "("where "|z-3|=t)`
`rArr 1lt|z-3|lt8//3`