If the equation
` ax^(2) + by^(2) + (a + b - 4) xy - ax - by - 20 = 0 `
represents a circle , then its radius is

Options

A. `(sqrt(21))/(2)`
B. `(sqrt(42))/(2)`
C. `2sqrt(21)`
D. `sqrt(22)`

Answer

To determine the radius of the circle represented by the equation

\[ ax^2 + by^2 + (a + b - 4)xy - ax - by - 20 = 0, \]

we need to ensure that this equation meets the conditions for a circle.

### Step 1: Identify the conditions for a circle
The general form of the equation of a circle is given by:

\[ x^2 + y^2 + 2gx + 2fy + c = 0, \]

where:
- The coefficients of \(x^2\) and \(y^2\) must be equal, i.e., \(a = b\).
- The coefficient of \(xy\) must be zero, i.e., \(a + b - 4 = 0\).

### Step 2: Set up the equations
From the conditions:
1. \(a = b\)
2. \(a + b - 4 = 0\)

Substituting \(b = a\) into the second equation gives:

\[ a + a - 4 = 0 \implies 2a - 4 = 0 \implies 2a = 4 \implies a = 2. \]

### Step 3: Find \(b\)
Since \(a = b\), we also have:

\[ b = 2. \]

### Step 4: Rewrite the equation
Now substituting \(a\) and \(b\) back into the original equation:

\[ 2x^2 + 2y^2 + (2 + 2 - 4)xy - 2x - 2y - 20 = 0 \implies 2x^2 + 2y^2 - 2x - 2y - 20 = 0. \]

Dividing the entire equation by 2 gives:

\[ x^2 + y^2 - x - y - 10 = 0. \]

### Step 5: Rearranging the equation
Rearranging this equation to standard form:

\[ x^2 - x + y^2 - y = 10. \]

### Step 6: Completing the square
To find the center and radius, we complete the square for both \(x\) and \(y\):

1. For \(x^2 - x\):
\[ x^2 - x = (x - \frac{1}{2})^2 - \frac{1}{4}. \]

2. For \(y^2 - y\):
\[ y^2 - y = (y - \frac{1}{2})^2 - \frac{1}{4}. \]

Substituting these back into the equation gives:

\[ (x - \frac{1}{2})^2 - \frac{1}{4} + (y - \frac{1}{2})^2 - \frac{1}{4} = 10. \]

Combining the constants:

\[ (x - \frac{1}{2})^2 + (y - \frac{1}{2})^2 = 10 + \frac{1}{4} + \frac{1}{4} = 10 + \frac{1}{2} = \frac{20}{2} + \frac{1}{2} = \frac{21}{2}. \]

### Step 7: Determine the radius
The standard form of a circle is:

\[ (x - h)^2 + (y - k)^2 = r^2, \]

where \(r\) is the radius. From our equation, we have:

\[ r^2 = \frac{21}{2} \implies r = \sqrt{\frac{21}{2}} = \frac{\sqrt{42}}{2}. \]

Thus, the radius of the circle is:

\[ \frac{\sqrt{42}}{2}. \]