To find the equation of the circumcircle of the curves given by the equations \(x^2 - 8x + 12 = 0\) and \(y^2 - 14y + 45 = 0\), we will follow these steps:
### Step 1: Factor the first equation \(x^2 - 8x + 12 = 0\)
We can factor this quadratic equation:
\[
x^2 - 8x + 12 = (x - 6)(x - 2) = 0
\]
From this, we find the roots:
\[
x = 6 \quad \text{and} \quad x = 2
\]
### Step 2: Factor the second equation \(y^2 - 14y + 45 = 0\)
Next, we factor the second quadratic equation:
\[
y^2 - 14y + 45 = (y - 9)(y - 5) = 0
\]
From this, we find the roots:
\[
y = 9 \quad \text{and} \quad y = 5
\]
### Step 3: Identify the points
The points of intersection (roots) from the equations are:
1. \( (2, 5) \)
2. \( (2, 9) \)
3. \( (6, 5) \)
4. \( (6, 9) \)
### Step 4: Determine the center and radius of the circumcircle
The circumcircle will have its center at the midpoint of the line segment connecting the points \( (2, 5) \) and \( (6, 9) \).
**Midpoint Calculation:**
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 6}{2}, \frac{5 + 9}{2} \right) = \left( 4, 7 \right)
\]
### Step 5: Calculate the radius
The radius is the distance from the center \( (4, 7) \) to one of the points, say \( (2, 5) \).
**Distance Calculation:**
\[
r = \sqrt{(4 - 2)^2 + (7 - 5)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
### Step 6: Write the equation of the circumcircle
The general equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Substituting \(h = 4\), \(k = 7\), and \(r = 2\sqrt{2}\):
\[
(x - 4)^2 + (y - 7)^2 = (2\sqrt{2})^2
\]
This simplifies to:
\[
(x - 4)^2 + (y - 7)^2 = 8
\]
### Step 7: Expand the equation
Expanding the equation gives:
\[
(x^2 - 8x + 16) + (y^2 - 14y + 49) = 8
\]
Combining terms results in:
\[
x^2 + y^2 - 8x - 14y + 57 = 0
\]
### Final Answer
Thus, the equation of the circumcircle is:
\[
x^2 + y^2 - 8x - 14y + 57 = 0
\]
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