To find the equation of the circle that passes through the points (2, 3), (2, 2), and (3, 2), we can follow these steps:
### Step 1: Write the general equation of a circle
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
### Step 2: Substitute the first point (2, 3) into the equation
Substituting \( (x, y) = (2, 3) \):
\[ 2^2 + 3^2 + 2g(2) + 2f(3) + c = 0 \]
This simplifies to:
\[ 4 + 9 + 4g + 6f + c = 0 \]
Thus, we have:
\[ 4g + 6f + c + 13 = 0 \]
This is our **Equation 1**.
### Step 3: Substitute the second point (2, 2) into the equation
Substituting \( (x, y) = (2, 2) \):
\[ 2^2 + 2^2 + 2g(2) + 2f(2) + c = 0 \]
This simplifies to:
\[ 4 + 4 + 4g + 4f + c = 0 \]
Thus, we have:
\[ 4g + 4f + c + 8 = 0 \]
This is our **Equation 2**.
### Step 4: Substitute the third point (3, 2) into the equation
Substituting \( (x, y) = (3, 2) \):
\[ 3^2 + 2^2 + 2g(3) + 2f(2) + c = 0 \]
This simplifies to:
\[ 9 + 4 + 6g + 4f + c = 0 \]
Thus, we have:
\[ 6g + 4f + c + 13 = 0 \]
This is our **Equation 3**.
### Step 5: Solve the equations
Now we have three equations:
1. \( 4g + 6f + c + 13 = 0 \) (Equation 1)
2. \( 4g + 4f + c + 8 = 0 \) (Equation 2)
3. \( 6g + 4f + c + 13 = 0 \) (Equation 3)
#### Step 5.1: Subtract Equation 2 from Equation 3
\[ (6g + 4f + c + 13) - (4g + 4f + c + 8) = 0 \]
This simplifies to:
\[ 2g + 5 = 0 \]
From this, we find:
\[ g = -\frac{5}{2} \]
#### Step 5.2: Subtract Equation 1 from Equation 2
\[ (4g + 4f + c + 8) - (4g + 6f + c + 13) = 0 \]
This simplifies to:
\[ -2f - 5 = 0 \]
From this, we find:
\[ f = -\frac{5}{2} \]
#### Step 5.3: Substitute values of g and f into Equation 1 to find c
Substituting \( g \) and \( f \) into Equation 1:
\[ 4(-\frac{5}{2}) + 6(-\frac{5}{2}) + c + 13 = 0 \]
This simplifies to:
\[ -10 - 15 + c + 13 = 0 \]
Thus:
\[ c - 12 = 0 \]
So, \( c = 12 \).
### Step 6: Write the equation of the circle
Now we can substitute the values of \( g \), \( f \), and \( c \) back into the general equation:
\[ x^2 + y^2 + 2(-\frac{5}{2})x + 2(-\frac{5}{2})y + 12 = 0 \]
This simplifies to:
\[ x^2 + y^2 - 5x - 5y + 12 = 0 \]
### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 5x - 5y + 12 = 0 \]
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