The equation `zbarz+(4-3i)z+(4+3i)barz+5=0` represents a circle of radius

Options

A. 5
B. `2sqrt(5)`
C. `5//2`
D. none of these

Answer

To solve the equation \( \overline{z}z + (4 - 3i)z + (4 + 3i)\overline{z} + 5 = 0 \) and find the radius of the circle it represents, we will follow these steps:

### Step 1: Substitute \( z \) with \( x + iy \)
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The conjugate of \( z \) is \( \overline{z} = x - iy \).

### Step 2: Rewrite the equation
Substituting \( z \) and \( \overline{z} \) into the equation, we have:
\[ \overline{z}z = (x - iy)(x + iy) = x^2 + y^2 \]
Now, substituting \( z \) and \( \overline{z} \) into the original equation:
\[ x^2 + y^2 + (4 - 3i)(x + iy) + (4 + 3i)(x - iy) + 5 = 0 \]

### Step 3: Expand the terms
Expanding \( (4 - 3i)(x + iy) \):
\[ = 4x + 4iy - 3ix - 3i^2y = 4x + 4iy - 3ix + 3y = 4x + 3y + (4y - 3x)i \]
Expanding \( (4 + 3i)(x - iy) \):
\[ = 4x - 4iy + 3ix - 3i^2y = 4x - 4iy + 3ix + 3y = 4x + 3y + (-4y + 3x)i \]

### Step 4: Combine the real and imaginary parts
Now, combine the real parts and the imaginary parts:
The real part:
\[ x^2 + y^2 + 4x + 3y + 4x + 3y + 5 = x^2 + y^2 + 8x + 6y + 5 \]
The imaginary part cancels out.

### Step 5: Set the equation to zero
Thus, we have:
\[ x^2 + y^2 + 8x + 6y + 5 = 0 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ x^2 + 8x + y^2 + 6y + 5 = 0 \]

### Step 7: Complete the square
To complete the square for \( x \) and \( y \):
1. For \( x^2 + 8x \):
\[ = (x + 4)^2 - 16 \]
2. For \( y^2 + 6y \):
\[ = (y + 3)^2 - 9 \]

Substituting back, we get:
\[ (x + 4)^2 - 16 + (y + 3)^2 - 9 + 5 = 0 \]
This simplifies to:
\[ (x + 4)^2 + (y + 3)^2 - 20 = 0 \]
or
\[ (x + 4)^2 + (y + 3)^2 = 20 \]

### Step 8: Identify the center and radius
This is the equation of a circle with center at \( (-4, -3) \) and radius \( r = \sqrt{20} = 2\sqrt{5} \).

### Final Answer
The radius of the circle represented by the given equation is \( 2\sqrt{5} \).

---