To find the equation of a circle passing through the points (5, 7), (6, 6), and (2, -2), we can follow these steps:
### Step 1: Set up the general equation of a circle
The general equation of a circle can be expressed as:
\[
x^2 + y^2 + Dx + Ey + F = 0
\]
where \(D\), \(E\), and \(F\) are constants that we need to determine.
### Step 2: Substitute the points into the circle's equation
We will substitute each of the three points into the circle's equation to create a system of equations.
1. For the point (5, 7):
\[
5^2 + 7^2 + 5D + 7E + F = 0 \implies 25 + 49 + 5D + 7E + F = 0 \implies 5D + 7E + F = -74 \quad \text{(Equation 1)}
\]
2. For the point (6, 6):
\[
6^2 + 6^2 + 6D + 6E + F = 0 \implies 36 + 36 + 6D + 6E + F = 0 \implies 6D + 6E + F = -72 \quad \text{(Equation 2)}
\]
3. For the point (2, -2):
\[
2^2 + (-2)^2 + 2D - 2E + F = 0 \implies 4 + 4 + 2D - 2E + F = 0 \implies 2D - 2E + F = -8 \quad \text{(Equation 3)}
\]
### Step 3: Solve the system of equations
We now have a system of three equations:
1. \(5D + 7E + F = -74\) (Equation 1)
2. \(6D + 6E + F = -72\) (Equation 2)
3. \(2D - 2E + F = -8\) (Equation 3)
We can eliminate \(F\) by subtracting Equation 1 from Equation 2:
\[
(6D + 6E + F) - (5D + 7E + F) = -72 + 74
\]
This simplifies to:
\[
D - E = 2 \quad \text{(Equation 4)}
\]
Next, we eliminate \(F\) again by subtracting Equation 3 from Equation 1:
\[
(5D + 7E + F) - (2D - 2E + F) = -74 + 8
\]
This simplifies to:
\[
3D + 9E = -66 \implies D + 3E = -22 \quad \text{(Equation 5)}
\]
### Step 4: Solve Equations 4 and 5
From Equation 4:
\[
D = E + 2
\]
Substituting \(D\) into Equation 5:
\[
(E + 2) + 3E = -22 \implies 4E + 2 = -22 \implies 4E = -24 \implies E = -6
\]
Now substituting \(E = -6\) back into Equation 4:
\[
D - (-6) = 2 \implies D + 6 = 2 \implies D = -4
\]
### Step 5: Find \(F\)
Now we can substitute \(D\) and \(E\) back into any of the original equations to find \(F\). Using Equation 1:
\[
5(-4) + 7(-6) + F = -74 \implies -20 - 42 + F = -74 \implies F = -74 + 62 \implies F = -12
\]
### Step 6: Write the equation of the circle
Now that we have \(D\), \(E\), and \(F\):
\[
D = -4, \quad E = -6, \quad F = -12
\]
The equation of the circle is:
\[
x^2 + y^2 - 4x - 6y - 12 = 0
\]
### Step 7: Find the center and radius
To find the center and radius, we can rewrite the equation in standard form:
\[
(x^2 - 4x) + (y^2 - 6y) = 12
\]
Completing the square:
\[
(x^2 - 4x + 4) + (y^2 - 6y + 9) = 12 + 4 + 9
\]
This simplifies to:
\[
(x - 2)^2 + (y - 3)^2 = 25
\]
Thus, the center of the circle is \((2, 3)\) and the radius is \(5\) (since \(\sqrt{25} = 5\)).
### Final Answer
The equation of the circle is:
\[
(x - 2)^2 + (y - 3)^2 = 25
\]
The center is \((2, 3)\) and the radius is \(5\).
