The equation of the circumcircle of the triangle formed by the lines whose combined equation is given by (x+y-4) (xy-2x-y+2)=0, is

Options

A. `x^(2)+y^(2)-5x-3y+8=0`
B. `x^(2)+y^(2)-3x-5y+8=0`
C. `x^(2)+y^(2)-3x-5y-8=0`
D. none of these

Explanation

To find the equation of the circumcircle of the triangle formed by the lines given by the combined equation \((x+y-4)(xy-2x-y+2)=0\), we can follow these steps:

### Step 1: Identify the Lines
The combined equation consists of two parts:
1. \(x + y - 4 = 0\)
2. \(xy - 2x - y + 2 = 0\)

We can rewrite the second equation:
\[ xy - 2x - y + 2 = 0 \implies y(x - 1) = 2x - 2 \implies y = \frac{2(x - 1)}{x - 1} \text{ (for } x \neq 1\text{)} \]
This gives us the line \(y = 2\) when \(x \neq 1\).

### Step 2: Find the Points of Intersection
Next, we need to find the points of intersection of these lines:
1. **Intersection of \(x + y - 4 = 0\) and \(y = 2\)**:
\[ x + 2 - 4 = 0 \implies x = 2 \implies \text{Point } A(2, 2) \]

2. **Intersection of \(x + y - 4 = 0\) and \(x = 1\)**:
\[ 1 + y - 4 = 0 \implies y = 3 \implies \text{Point } B(1, 3) \]

3. **Intersection of \(y = 2\) and \(x = 1\)**:
\[ \text{Point } C(1, 2) \]

### Step 3: Identify the Triangle Vertices
The vertices of the triangle are:
- \(A(2, 2)\)
- \(B(1, 3)\)
- \(C(1, 2)\)

### Step 4: Find the Circumcircle
To find the circumcircle, we can use the fact that if one of the angles of the triangle is \(90^\circ\), the circumcircle's diameter is the hypotenuse.

Here, the triangle formed by points \(A\), \(B\), and \(C\) has a right angle at \(C\).

### Step 5: Equation of the Circle
The diameter endpoints are \(A(2, 2)\) and \(B(1, 3)\). The equation of the circle with diameter endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]
Substituting the points \(A(2, 2)\) and \(B(1, 3)\):
\[ (x - 2)(x - 1) + (y - 2)(y - 3) = 0 \]

### Step 6: Expand the Equation
Expanding the equation:
\[ (x^2 - 3x + 2) + (y^2 - 5y + 6) = 0 \]
Combining like terms:
\[ x^2 + y^2 - 3x - 5y + 8 = 0 \]

### Final Equation
Thus, the equation of the circumcircle is:
\[ x^2 + y^2 - 3x - 5y + 8 = 0 \]