Find the equation of a circle passing through the points (5,7), (6,6) and (2,-1). Also, find its centre and radius.

Answer

To find the equation of a circle passing through the points (5,7), (6,6), and (2,-1), we will use the general equation of a circle:

\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

where \((g, f)\) are the coordinates of the center of the circle, and \(c\) is a constant.

### Step 1: Substitute the points into the circle equation

1. **For point (5, 7)**:
\[ 5^2 + 7^2 + 2g(5) + 2f(7) + c = 0 \]
\[ 25 + 49 + 10g + 14f + c = 0 \]
\[ 10g + 14f + c = -74 \quad \text{(Equation 1)} \]

2. **For point (6, 6)**:
\[ 6^2 + 6^2 + 2g(6) + 2f(6) + c = 0 \]
\[ 36 + 36 + 12g + 12f + c = 0 \]
\[ 12g + 12f + c = -72 \quad \text{(Equation 2)} \]

3. **For point (2, -1)**:
\[ 2^2 + (-1)^2 + 2g(2) + 2f(-1) + c = 0 \]
\[ 4 + 1 + 4g - 2f + c = 0 \]
\[ 4g - 2f + c = -5 \quad \text{(Equation 3)} \]

### Step 2: Solve the system of equations

Now we have three equations:
1. \(10g + 14f + c = -74\) (Equation 1)
2. \(12g + 12f + c = -72\) (Equation 2)
3. \(4g - 2f + c = -5\) (Equation 3)

**Subtract Equation 1 from Equation 2**:
\[ (12g + 12f + c) - (10g + 14f + c) = -72 + 74 \]
\[ 2g - 2f = 2 \implies g - f = 1 \quad \text{(Equation 4)} \]

**Subtract Equation 3 from Equation 1**:
\[ (10g + 14f + c) - (4g - 2f + c) = -74 + 5 \]
\[ 6g + 16f = -69 \quad \text{(Equation 5)} \]

### Step 3: Substitute \(f\) from Equation 4 into Equation 5

From Equation 4:
\[ f = g - 1 \]

Substituting into Equation 5:
\[ 6g + 16(g - 1) = -69 \]
\[ 6g + 16g - 16 = -69 \]
\[ 22g - 16 = -69 \]
\[ 22g = -53 \implies g = -\frac{53}{22} \]

### Step 4: Find \(f\) using \(g\)

Substituting \(g\) back into Equation 4:
\[ f = -\frac{53}{22} - 1 = -\frac{53}{22} - \frac{22}{22} = -\frac{75}{22} \]

### Step 5: Find \(c\)

Substituting \(g\) and \(f\) into Equation 1:
\[ 10(-\frac{53}{22}) + 14(-\frac{75}{22}) + c = -74 \]
\[ -\frac{530}{22} - \frac{1050}{22} + c = -74 \]
\[ -\frac{1580}{22} + c = -\frac{1628}{22} \]
\[ c = -\frac{1628}{22} + \frac{1580}{22} = -\frac{48}{22} = -\frac{24}{11} \]

### Step 6: Write the equation of the circle

Now we have:
- \(g = -\frac{53}{22}\)
- \(f = -\frac{75}{22}\)
- \(c = -\frac{24}{11}\)

The equation of the circle is:
\[ x^2 + y^2 - \frac{53}{11}x - \frac{75}{11}y - \frac{24}{11} = 0 \]

### Step 7: Find the center and radius

The center \((h, k)\) of the circle is:
\[ \left(-g, -f\right) = \left(\frac{53}{22}, \frac{75}{22}\right) \]

The radius \(r\) is given by:
\[ r = \sqrt{g^2 + f^2 - c} \]
Calculating:
\[ r = \sqrt{\left(-\frac{53}{22}\right)^2 + \left(-\frac{75}{22}\right)^2 - \left(-\frac{24}{11}\right)} \]
\[ = \sqrt{\frac{2809}{484} + \frac{5625}{484} + \frac{48}{11}} \]
\[ = \sqrt{\frac{2809 + 5625 + 192}{484}} = \sqrt{\frac{8626}{484}} = \sqrt{\frac{4313}{242}} \approx 4.31 \]

### Final Results:
- **Equation of the Circle**:
\[ x^2 + y^2 - \frac{53}{11}x - \frac{75}{11}y - \frac{24}{11} = 0 \]
- **Center**:
\[ \left(\frac{53}{22}, \frac{75}{22}\right) \]
- **Radius**:
\[ \sqrt{\frac{4313}{242}} \approx 4.31 \]