The set of values of k for which the equation `zbarz+(-3+4i)barz-(3+4i)z+k=0`
represents a circle, is

Answer

To solve the equation \( \overline{z}z + (-3 + 4i)\overline{z} - (3 + 4i)z + k = 0 \) and find the set of values of \( k \) for which it represents a circle, we can follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ \overline{z}z + (-3 + 4i)\overline{z} - (3 + 4i)z + k = 0 \]
We can rewrite it as:
\[ z\overline{z} + (-3 + 4i)\overline{z} - (3 + 4i)z + k = 0 \]

### Step 2: Identify the terms
In the equation, we can identify:
- \( z\overline{z} \) is the modulus squared of \( z \).
- The terms involving \( z \) and \( \overline{z} \) can be grouped.

### Step 3: Compare with the circle equation
The general form of the equation of a circle in complex numbers is:
\[ z\overline{z} + \alpha \overline{z} + \overline{\alpha} z + c = 0 \]
where \( \alpha \) is a complex number representing the center of the circle, and \( c \) is a constant.

### Step 4: Identify \( \alpha \) and \( c \)
From our equation, we can see:
- \( \alpha = -3 + 4i \)
- \( \overline{\alpha} = -3 - 4i \)
- \( c = k \)

### Step 5: Calculate the modulus of \( \alpha \)
To find the radius of the circle, we need to calculate the modulus of \( \alpha \):
\[ |\alpha| = | -3 + 4i | = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 6: Determine the radius condition
The radius \( r \) of the circle is given by:
\[ r = \sqrt{|\alpha|^2 - c} = \sqrt{25 - k} \]
For the equation to represent a circle, the radius must be greater than 0:
\[ \sqrt{25 - k} > 0 \]

### Step 7: Solve the inequality
Squaring both sides, we get:
\[ 25 - k > 0 \]
This simplifies to:
\[ k < 25 \]

### Step 8: Conclusion
Thus, the set of values of \( k \) for which the equation represents a circle is:
\[ (-\infty, 25) \]