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

Explanation

To solve the problem, we need to analyze the given inequality involving the complex number \( z \) and its distance from the center of the circles. Let's break it down step by step.

### Step 1: Understand the circles
The circles \( C_1 \) and \( C_2 \) are concentric with the center at \( (3, 0) \) on the Argand plane. The radius of \( C_1 \) is \( 1 \) and the radius of \( C_2 \) is \( \frac{8}{3} \).

### Step 2: Rewrite the inequality
We are given the inequality:
\[ \log_{1/3}\left(\frac{|z-3|^2 + 2}{11|z-3| - 2}\right) > 1 \]
Since the logarithm base \( \frac{1}{3} \) is decreasing, we can rewrite the inequality:
\[ \frac{|z-3|^2 + 2}{11|z-3| - 2} < \frac{1}{3} \]

### Step 3: Cross-multiply
To eliminate the fraction, we cross-multiply:
\[ 3(|z-3|^2 + 2) < 11|z-3| - 2 \]
Expanding this gives:
\[ 3|z-3|^2 + 6 < 11|z-3| - 2 \]
Rearranging terms, we have:
\[ 3|z-3|^2 - 11|z-3| + 8 < 0 \]

### Step 4: Let \( t = |z-3| \)
Let \( t = |z-3| \). The inequality becomes:
\[ 3t^2 - 11t + 8 < 0 \]

### Step 5: Solve the quadratic inequality
To find the roots of the quadratic equation \( 3t^2 - 11t + 8 = 0 \), we use the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{11 \pm \sqrt{(-11)^2 - 4 \cdot 3 \cdot 8}}{2 \cdot 3} \]
Calculating the discriminant:
\[ 121 - 96 = 25 \]
Thus, the roots are:
\[ t = \frac{11 \pm 5}{6} \]
Calculating the two roots:
\[ t_1 = \frac{16}{6} = \frac{8}{3}, \quad t_2 = \frac{6}{6} = 1 \]

### Step 6: Analyze the intervals
The quadratic \( 3t^2 - 11t + 8 \) opens upwards (since the coefficient of \( t^2 \) is positive). The inequality \( 3t^2 - 11t + 8 < 0 \) holds between the roots:
\[ 1 < t < \frac{8}{3} \]
This translates back to:
\[ 1 < |z - 3| < \frac{8}{3} \]

### Step 7: Interpret the result
The condition \( |z - 3| > 1 \) means that \( z \) lies outside the circle \( C_1 \) (radius 1), and \( |z - 3| < \frac{8}{3} \) means that \( z \) lies inside the circle \( C_2 \) (radius \( \frac{8}{3} \)).

### Conclusion
Thus, the correct option is:
(a) \( z \) lies outside \( C_1 \) but inside \( C_2 \).