Find the centre and radius of the circle which passes through lie points (7,5), (6, - 2), (-1 , -1 )

Answer

To find the center and radius of the circle that passes through the points (7, 5), (6, -2), and (-1, -1), we can use the general equation of a circle:

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

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

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

2. **For the point (6, -2)**:
\[ 6^2 + (-2)^2 + 2g(6) + 2f(-2) + c = 0 \]
\[ 36 + 4 + 12g - 4f + c = 0 \quad \text{(Equation 2)} \]
Simplifying, we get:
\[ 12g - 4f + c + 40 = 0 \]

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

### Step 2: Set up the system of equations

From the above, we have the following equations:

1. \( 14g + 10f + c + 74 = 0 \) (Equation 1)
2. \( 12g - 4f + c + 40 = 0 \) (Equation 2)
3. \( -2g - 2f + c + 2 = 0 \) (Equation 3)

### Step 3: Eliminate \(c\)

We can eliminate \(c\) from these equations by subtracting them.

1. **Subtract Equation 2 from Equation 1**:
\[ (14g + 10f + c + 74) - (12g - 4f + c + 40) = 0 \]
This simplifies to:
\[ 2g + 14f + 34 = 0 \quad \text{(Equation 4)} \]

2. **Subtract Equation 3 from Equation 1**:
\[ (14g + 10f + c + 74) - (-2g - 2f + c + 2) = 0 \]
This simplifies to:
\[ 16g + 12f + 72 = 0 \quad \text{(Equation 5)} \]

### Step 4: Solve the system of equations

Now we have two equations (Equation 4 and Equation 5):

1. \( 2g + 14f + 34 = 0 \)
2. \( 16g + 12f + 72 = 0 \)

We can solve these equations simultaneously.

1. **From Equation 4**:
\[ 2g = -14f - 34 \quad \Rightarrow \quad g = -7f - 17 \]

2. **Substituting \(g\) into Equation 5**:
\[ 16(-7f - 17) + 12f + 72 = 0 \]
This simplifies to:
\[ -112f - 272 + 12f + 72 = 0 \]
Combining like terms:
\[ -100f - 200 = 0 \quad \Rightarrow \quad f = -2 \]

3. **Substituting \(f\) back to find \(g\)**:
\[ g = -7(-2) - 17 = 14 - 17 = -3 \]

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

Now, we can substitute \(g\) and \(f\) back into any of the original equations to find \(c\). Using Equation 3:
\[ -2(-3) - 2(-2) + c + 2 = 0 \]
\[ 6 + 4 + c + 2 = 0 \quad \Rightarrow \quad c + 12 = 0 \quad \Rightarrow \quad c = -12 \]

### Step 6: Find the center and radius

The center of the circle is given by \((-g, -f)\):
\[ \text{Center} = (3, 2) \]

The radius \(r\) is given by:
\[ r = \sqrt{g^2 + f^2 - c} = \sqrt{(-3)^2 + (-2)^2 - (-12)} = \sqrt{9 + 4 + 12} = \sqrt{25} = 5 \]

### Final Answer
- **Center**: (3, 2)
- **Radius**: 5