Question

How do I find the circle equation if I have three points lying on the circle

Answer 1

To find the equation of a circle given three points that lie on it, you can follow these steps:

### Step 1: Identify the Points
Let the three points be \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).

### Step 2: Set Up the General Equation of the Circle
The general equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \( (h, k) \) is the center and \( r \) is the radius. By expanding, it can also be stated as:
\[
x^2 + y^2 + Dx + Ey + F = 0
\]
where \( D, E, F \) are constants.

### Step 3: Substitute the Points into the Circle Equation
For each of the three points, substitute their coordinates into the circle equation:

1. For point \( A(x_1, y_1) \):
\[
x_1^2 + y_1^2 + Dx_1 + Ey_1 + F = 0 \quad \text{(1)}
\]

2. For point \( B(x_2, y_2) \):
\[
x_2^2 + y_2^2 + Dx_2 + Ey_2 + F = 0 \quad \text{(2)}
\]

3. For point \( C(x_3, y_3) \):
\[
x_3^2 + y_3^2 + Dx_3 + Ey_3 + F = 0 \quad \text{(3)}
\]

### Step 4: Create a System of Linear Equations
From these three equations, you can form a system of equations in terms of \( D, E, \) and \( F \). Rearranging gives you:
\[
Dx_1 + Ey_1 + F = - (x_1^2 + y_1^2) \quad \text{(from 1)}
\]
\[
Dx_2 + Ey_2 + F = - (x_2^2 + y_2^2) \quad \text{(from 2)}
\]
\[
Dx_3 + Ey_3 + F = - (x_3^2 + y_3^2) \quad \text{(from 3)}
\]

### Step 5: Solve the System of Equations
You can represent the equations in matrix form:
\[
\begin{bmatrix}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{bmatrix}
\begin{bmatrix}
D \\
E \\
F
\end{bmatrix}
=
\begin{bmatrix}
-(x_1^2 + y_1^2) \\
-(x_2^2 + y_2^2) \\
-(x_3^2 + y_3^2)
\end{bmatrix}
\]

You can use methods such as substitution, elimination, or matrix inversion (if applicable) to find \( D, E, \) and \( F \).

### Step 6: Write the Circle Equation
Once you have \( D, E, \) and \( F \), substitute them back into the general equation of the circle:
\[
x^2 + y^2 + Dx + Ey + F = 0
\]

### Step 7: Convert to Standard Form (Optional)
If you’d like to express the circle in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \), you can complete the square on the \( x \) and \( y \) terms.

### Example
Suppose points A(1, 2), B(4, 6), and C(5, 2) are given. Substitute these points into the equations, solve for \( D, E, \) and \( F \), and express the resulting equation.

By following these steps, you can derive the equation of a circle that passes through any three given points.