If ` log_(sqrt(3))((| z| ^(2) - | z| + 1)/( 2 + | z|)) lt 2 ` then the locus of z is

Options

A. z = 5
B. `| z| lt 5 `
C. ` | z| gt 5 `
D. none of these

Answer

To solve the problem, we need to analyze the inequality given by the logarithmic expression. The question states:

If \( \log_{\sqrt{3}}\left(\frac{|z|^2 - |z| + 1}{2 + |z|}\right) < 2 \), then find the locus of \( z \).

### Step-by-Step Solution:

1. **Convert the logarithmic inequality to exponential form**:
\[ \frac{|z|^2 - |z| + 1}{2 + |z|} < (\sqrt{3})^2 \]
Since \( (\sqrt{3})^2 = 3 \), we rewrite the inequality as:
\[ \frac{|z|^2 - |z| + 1}{2 + |z|} < 3 \]

2. **Cross-multiply to eliminate the fraction**:
\[ |z|^2 - |z| + 1 < 3(2 + |z|) \]
Expanding the right-hand side:
\[ |z|^2 - |z| + 1 < 6 + 3|z| \]

3. **Rearranging the inequality**:
\[ |z|^2 - |z| - 3|z| + 1 - 6 < 0 \]
Simplifying this gives:
\[ |z|^2 - 4|z| - 5 < 0 \]

4. **Factor the quadratic expression**:
We need to factor \( |z|^2 - 4|z| - 5 \):
\[ |z|^2 - 4|z| - 5 = (|z| - 5)(|z| + 1) \]

5. **Set up the inequality**:
The inequality now reads:
\[ (|z| - 5)(|z| + 1) < 0 \]

6. **Analyze the roots**:
The roots of the equation \( (|z| - 5)(|z| + 1) = 0 \) are:
- \( |z| = 5 \)
- \( |z| = -1 \) (which is not possible since \( |z| \) is always non-negative)

7. **Determine the intervals**:
We need to find where the product \( (|z| - 5)(|z| + 1) < 0 \):
- The critical points are \( |z| = 5 \) and \( |z| = -1 \).
- The intervals to test are \( (-\infty, -1) \), \( (-1, 5) \), and \( (5, \infty) \).

Since \( |z| \) cannot be negative, we only consider \( |z| \geq 0 \):
- For \( |z| \in (0, 5) \): The expression is negative (since \( |z| - 5 < 0 \) and \( |z| + 1 > 0 \)).
- For \( |z| = 5 \): The expression is zero.
- For \( |z| > 5 \): The expression is positive.

8. **Conclusion**:
The inequality \( (|z| - 5)(|z| + 1) < 0 \) holds for:
\[ 0 < |z| < 5 \]
Thus, the locus of \( z \) is a circle of radius 5 centered at the origin, excluding the boundary.

### Final Answer:
The locus of \( z \) is \( |z| < 5 \).