If $|z+1|=\sqrt{2}|z-1|$, where $z=x+iy, x,y\in R$, then the locus described by the point z in the Argand plane is a :

Question

Accepted Answer

To determine the locus described by the point $ z $ in the Argand plane given the condition $ |z+1| = \sqrt{2}|z-1| $, we need to analyze the equation geometrically.

### Explanation for Option 2: Circle

1. **Geometric Interpretation**:
   - The expression $ |z+1| $ represents the distance from the point $ z $ to the point $-1 + 0i$ in the complex plane.
   - Similarly, $ |z-1| $ represents the distance from the point $ z $ to the point $ 1 + 0i $.

2. **Given Condition**:
   - The equation $ |z+1| = \sqrt{2}|z-1| $ implies that the distance from $ z $ to $-1$ is $\sqrt{2}$ times the distance from $ z $ to $ 1 $.

3. **Locus Derivation**:
   - This condition describes a set of points $ z $ such that the ratio of distances to two fixed points is constant. This is a well-known geometric property of a circle.
   - Specifically, the locus of points $ z $ such that the ratio of distances to two fixed points is constant (and not equal to 1) is a circle.

4. **Conclusion**:
   - Therefore, the locus of $ z $ satisfying the given condition is a circle.

### Explanation for Incorrect Options:

1. **Option 1: Straight Line**:
   - A straight line would imply a linear relationship between the distances, typically represented by an equation of the form $ |z - a| = |z - b| $, which describes the perpendicular bisector of the segment joining $ a $ and $ b $. The given condition does not fit this form.

2. **Option 3: Parabola**:
   - A parabola is the locus of points equidistant from a fixed point (focus) and a fixed line (directrix). The given condition involves two fixed points and a constant ratio of distances, which does not correspond to the definition of a parabola.

3. **Option 4: Ellipse**:
   - An ellipse is defined as the locus 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 correct interpretation of the given condition $ |z+1| = \sqrt{2}|z-1| $ is that it describes a circle in the Argand plane, making Option 2 the correct answer.