The equation of the circumcricle of the `x^(2)-8x+12=0 and y^(2)-14y+45=0` is

Options

A. `x^(2)+y^(2)-4x-7y+57=0`
B. `x^(2)+y^(2)-8x-14y+57=0`
C. `x^(2)+y^(2)-8x-|14y+5|=0`
D. `2x^(2)+y^(2)-8x-14y+57=0`

Answer

To find the equation of the circumcircle of the lines represented by the equations \(x^2 - 8x + 12 = 0\) and \(y^2 - 14y + 45 = 0\), we will follow these steps:

### Step 1: Solve the equations for x and y
1. **Solve \(x^2 - 8x + 12 = 0\)**:
- This can be factored as \((x - 6)(x - 2) = 0\).
- Therefore, \(x = 6\) and \(x = 2\).

2. **Solve \(y^2 - 14y + 45 = 0\)**:
- This can be factored as \((y - 9)(y - 5) = 0\).
- Therefore, \(y = 9\) and \(y = 5\).

### Step 2: Identify the points of intersection
From the solutions, we have four points of intersection:
- \((2, 5)\)
- \((2, 9)\)
- \((6, 5)\)
- \((6, 9)\)

### Step 3: Find the center of the circumcircle
The center of the circumcircle can be found by calculating the midpoint of the diagonal points:
- Midpoint \(M\) of points \((2, 5)\) and \((6, 9)\):
\[ M = \left(\frac{2 + 6}{2}, \frac{5 + 9}{2}\right) = \left(4, 7\right) \]

### Step 4: Calculate the radius of the circumcircle
The radius can be calculated using the distance formula between the center and one of the points, say \((2, 5)\):
\[ \text{Radius} = \sqrt{(6 - 2)^2 + (9 - 5)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
Thus, the radius \(r = 2\sqrt{2}\).

### Step 5: Write the equation of the circumcircle
The standard form of the equation of a circle is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h = 4\), \(k = 7\), and \(r = 2\sqrt{2}\):
\[ (x - 4)^2 + (y - 7)^2 = (2\sqrt{2})^2 \]
\[ (x - 4)^2 + (y - 7)^2 = 8 \]

### Step 6: Expand the equation
Expanding the equation:
\[ (x^2 - 8x + 16) + (y^2 - 14y + 49) = 8 \]
Combine like terms:
\[ x^2 + y^2 - 8x - 14y + 65 - 8 = 0 \]
\[ x^2 + y^2 - 8x - 14y + 57 = 0 \]

### Final Answer
The equation of the circumcircle is:
\[ x^2 + y^2 - 8x - 14y + 57 = 0 \]