If $z=x+iy$ and $|\frac{z-2}{z+2}|=\frac{\pi}{6},$ then locus of z is:

Question

Accepted Answer

To determine the locus of $ z = x + iy $ given the condition $ \left|\frac{z-2}{z+2}\right| = \frac{\pi}{6} $, we need to analyze the expression and understand its geometric interpretation.

### Explanation for Option 2: A Circle

The expression $ \left|\frac{z-2}{z+2}\right| = \frac{\pi}{6} $ represents a condition on the complex number $ z $. In complex analysis, the expression $ \left|\frac{z-a}{z-b}\right| = k $ (where $ a $ and $ b $ are complex numbers and $ k $ is a positive real number) describes a circle if $ k \neq 1 $.

1. **Geometric Interpretation**: 
   - The expression $ \left|\frac{z-2}{z+2}\right| = \frac{\pi}{6} $ can be interpreted as the set of points $ z $ such that the ratio of the distances from $ z $ to the points $ 2 $ and $ -2 $ is constant. Specifically, the distance from $ z $ to $ 2 $ divided by the distance from $ z $ to $ -2 $ is $ \frac{\pi}{6} $.
   - This is a classic definition of an Apollonian circle, which is a circle that is the locus of points for which the ratio of distances to two fixed points is constant.

2. **Why it is a Circle**:
   - Since $ \frac{\pi}{6} \neq 1 $, the locus is indeed a circle. If $ k = 1 $, the locus would be the perpendicular bisector of the line segment joining $ 2 $ and $ -2 $, which is a line. However, since $ k = \frac{\pi}{6} $, the locus is a circle.

### Explanation for Why Other Options are Incorrect

1. **Option 1: A Line**:
   - As mentioned, the locus would be a line only if $ \left|\frac{z-2}{z+2}\right| = 1 $. Since $ \frac{\pi}{6} \neq 1 $, the locus cannot be a line.

2. **Option 3: A Parabola**:
   - A parabola is defined as the set of points equidistant from a point (focus) and a line (directrix). The given condition does not fit this definition, as it involves a ratio of distances to two points, not a point and a line.

3. **Option 4: An Ellipse**:
   - An ellipse is defined as the set of points for which the sum of distances to two fixed points is constant. The given condition involves a ratio of distances, not a sum, so it does not describe an ellipse.

In conclusion, the condition $ \left|\frac{z-2}{z+2}\right| = \frac{\pi}{6} $ describes a circle, making Option 2 the correct answer.