find the equation of the circle which passes through the points `(20,3), (19, 8)and (2, -9)` find its centre and radius

Answer

To find the equation of the circle that passes through the points (20, 3), (19, 8), and (2, -9), we will follow these steps:

### Step 1: Write the general equation of the circle
The general equation of a circle can be expressed as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where (−g, −f) is the center of the circle and the radius can be calculated using the formula \( \sqrt{g^2 + f^2 - c} \).

### Step 2: Substitute the points into the equation
We will substitute each of the three points into the general equation to create a system of equations.

1. For the point (20, 3):
\[ 20^2 + 3^2 + 2g(20) + 2f(3) + c = 0 \]
This simplifies to:
\[ 400 + 9 + 40g + 6f + c = 0 \implies 40g + 6f + c = -409 \quad \text{(Equation 1)} \]

2. For the point (19, 8):
\[ 19^2 + 8^2 + 2g(19) + 2f(8) + c = 0 \]
This simplifies to:
\[ 361 + 64 + 38g + 16f + c = 0 \implies 38g + 16f + c = -425 \quad \text{(Equation 2)} \]

3. For the point (2, -9):
\[ 2^2 + (-9)^2 + 2g(2) + 2f(-9) + c = 0 \]
This simplifies to:
\[ 4 + 81 + 4g - 18f + c = 0 \implies 4g - 18f + c = -85 \quad \text{(Equation 3)} \]

### Step 3: Solve the system of equations
Now we have a system of three equations:
1. \( 40g + 6f + c = -409 \)
2. \( 38g + 16f + c = -425 \)
3. \( 4g - 18f + c = -85 \)

We can eliminate \( c \) by subtracting Equation 1 from Equations 2 and 3.

#### Subtract Equation 1 from Equation 2:
\[ (38g + 16f + c) - (40g + 6f + c) = -425 + 409 \]
This simplifies to:
\[ -2g + 10f = -16 \implies 2g - 10f = 16 \quad \text{(Equation 4)} \]

#### Subtract Equation 1 from Equation 3:
\[ (4g - 18f + c) - (40g + 6f + c) = -85 + 409 \]
This simplifies to:
\[ -36g - 24f = 324 \implies 3g + 2f = -27 \quad \text{(Equation 5)} \]

### Step 4: Solve Equations 4 and 5
Now we have two equations:
1. \( 2g - 10f = 16 \)
2. \( 3g + 2f = -27 \)

From Equation 4, we can express \( g \) in terms of \( f \):
\[ g = 5f + 8 \]

Substituting this into Equation 5:
\[ 3(5f + 8) + 2f = -27 \]
This simplifies to:
\[ 15f + 24 + 2f = -27 \implies 17f = -51 \implies f = -3 \]

Now substituting \( f = -3 \) back into the expression for \( g \):
\[ g = 5(-3) + 8 = -15 + 8 = -7 \]

### Step 5: Find \( c \)
Substituting \( g \) and \( f \) back into Equation 1 to find \( c \):
\[ 40(-7) + 6(-3) + c = -409 \]
This simplifies to:
\[ -280 - 18 + c = -409 \implies c = -409 + 298 = -111 \]

### Step 6: Write the equation of the circle
Now we have \( g = -7 \), \( f = -3 \), and \( c = -111 \). The equation of the circle is:
\[ x^2 + y^2 - 14x - 6y - 111 = 0 \]

### Step 7: Find the center and radius
The center of the circle is given by \( (-g, -f) \):
\[ \text{Center} = (7, 3) \]

The radius is given by:
\[ \text{Radius} = \sqrt{g^2 + f^2 - c} = \sqrt{(-7)^2 + (-3)^2 - (-111)} = \sqrt{49 + 9 + 111} = \sqrt{169} = 13 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 14x - 6y - 111 = 0 \]
The center is \( (7, 3) \) and the radius is \( 13 \).