If `z_(1) and z_(2)` are two fixed points in the Argand plane, then find the locus of a point z in each of the following
`|z-z_(1)|= k|z-z_(2)|, k in R^(+), k ne 1`

Answer

To find the locus of the point \( z \) in the Argand plane given the equation \( |z - z_1| = k |z - z_2| \) where \( k \in \mathbb{R}^+ \) and \( k \neq 1 \), we can follow these steps:

### Step 1: Understanding the Equation
The equation \( |z - z_1| = k |z - z_2| \) means that the distance from the point \( z \) to the point \( z_1 \) is \( k \) times the distance from \( z \) to the point \( z_2 \).

### Step 2: Squaring Both Sides
To eliminate the modulus, we square both sides:
\[ |z - z_1|^2 = k^2 |z - z_2|^2 \]

### Step 3: Expanding Both Sides
Using the property \( |a|^2 = a \overline{a} \), we can expand both sides:
\[ (z - z_1)(\overline{z} - \overline{z_1}) = k^2 (z - z_2)(\overline{z} - \overline{z_2}) \]
This results in:
\[ (z \overline{z} - z z_1^* - z_1 z^* + z_1 z_1^*) = k^2 (z \overline{z} - z z_2^* - z_2 z^* + z_2 z_2^*) \]

### Step 4: Rearranging the Equation
Rearranging gives us:
\[ z \overline{z} - z z_1^* - z_1 z^* + z_1 z_1^* - k^2 z \overline{z} + k^2 z z_2^* + k^2 z_2 z^* - k^2 z_2 z_2^* = 0 \]
Combining like terms, we have:
\[ (1 - k^2) z \overline{z} - z z_1^* + k^2 z z_2^* - z_1 z^* + k^2 z_2 z^* + z_1 z_1^* - k^2 z_2 z_2^* = 0 \]

### Step 5: Dividing by \( 1 - k^2 \)
Since \( k \neq 1 \), we can divide the entire equation by \( 1 - k^2 \):
\[ z \overline{z} - \frac{z z_1^*}{1 - k^2} + \frac{k^2 z z_2^*}{1 - k^2} - \frac{z_1 z^*}{1 - k^2} + \frac{k^2 z_2 z^*}{1 - k^2} + \frac{z_1 z_1^* - k^2 z_2 z_2^*}{1 - k^2} = 0 \]

### Step 6: Identifying the Circle's Center and Radius
This equation can be compared to the general equation of a circle:
\[ |z - z_0|^2 = r^2 \]
From our rearranged equation, we can identify the center \( z_0 \) and radius \( r \):
- The center \( z_0 = \frac{z_1 - k^2 z_2}{1 - k^2} \)
- The radius \( r = \frac{k |z_1 - z_2|}{|1 - k^2|} \)

### Conclusion
Thus, the locus of the point \( z \) is a circle with center \( \frac{z_1 - k^2 z_2}{1 - k^2} \) and radius \( \frac{k |z_1 - z_2|}{|1 - k^2|} \).