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:
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For point \( A(x_1, y_1) \): \[ x_1^2 + y_1^2 + Dx_1 + Ey_1 + F = 0 \quad \text{(1)} \]
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For point \( B(x_2, y_2) \): \[ x_2^2 + y_2^2 + Dx_2 + Ey_2 + F = 0 \quad \text{(2)} \]
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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.