The equation `(x+y=6)(xy-3x-y+3=0)` represents the sides of a triangle then the equation of the circumcircle of the triangle is

Options

A. `x^(2)+y^(2)-5x-9y+20=0`
B. `x^(2)+y^(2)-4x-8y+18=0`
C. `x^(2)+y^(2)-3x-5y+8=0`
D. `x^(2)+y^(2)+2x-3y-1=0`

Answer

To find the equation of the circumcircle of the triangle formed by the lines \(x + y = 6\) and \(xy - 3x - y + 3 = 0\), we will follow these steps:

### Step 1: Identify the lines
We have two equations:
1. \(x + y = 6\)
2. \(xy - 3x - y + 3 = 0\)

### Step 2: Solve for the points of intersection
To find the vertices of the triangle, we need to find the points where these lines intersect.

From the first equation, we can express \(y\) in terms of \(x\):
\[ y = 6 - x \]

Now, substitute this expression for \(y\) into the second equation:
\[ x(6 - x) - 3x - (6 - x) + 3 = 0 \]
Expanding this:
\[ 6x - x^2 - 3x - 6 + x + 3 = 0 \]
Combining like terms:
\[ -x^2 + 4x - 3 = 0 \]
Multiplying through by -1:
\[ x^2 - 4x + 3 = 0 \]
Factoring:
\[ (x - 1)(x - 3) = 0 \]
Thus, \(x = 1\) or \(x = 3\).

### Step 3: Find corresponding \(y\) values
For \(x = 1\):
\[ y = 6 - 1 = 5 \quad \Rightarrow \quad (1, 5) \]

For \(x = 3\):
\[ y = 6 - 3 = 3 \quad \Rightarrow \quad (3, 3) \]

### Step 4: Identify the third vertex
The third vertex is where the line \(x = 1\) intersects the line \(y = 3\):
\[ (1, 3) \]

### Step 5: List the vertices of the triangle
The vertices of the triangle are:
1. \(A(1, 5)\)
2. \(B(3, 3)\)
3. \(C(1, 3)\)

### Step 6: Find the circumcircle
The circumcircle of a triangle can be found using the formula:
\[ (x - x_1)(x - x_2)(y - y_1)(y - y_2) = 0 \]
Where \((x_1, y_1)\), \((x_2, y_2)\) are the coordinates of the vertices.

Using the vertices \(A(1, 5)\), \(B(3, 3)\), and \(C(1, 3)\), we can derive the circumcircle equation.

### Step 7: Use the general form of the circumcircle
The general equation of a circle is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Where \((h, k)\) is the center and \(r\) is the radius.

To find the circumcircle, we can use the determinant method or find the circumcenter and radius, but for simplicity, we will derive it directly from the points.

### Step 8: Final equation of the circumcircle
The circumcircle can be expressed as:
\[ (x - 1)(x - 3) + (y - 5)(y - 3) = 0 \]
Expanding this gives:
\[ x^2 - 4x + 3 + y^2 - 8y + 15 = 0 \]
Combining terms:
\[ x^2 + y^2 - 4x - 8y + 18 = 0 \]

### Final Answer
The equation of the circumcircle of the triangle is:
\[ x^2 + y^2 - 4x - 8y + 18 = 0 \]