To find the general form of the equation of a circle that passes through three points $A(2, 3)$, $B(6, 1)$, and $C(4, -3)$, we follow these steps:

Step 1: General Form of the Circle

The general equation of a circle is: $x^2 + y^2 + Dx + Ey + F = 0$ where $D$, $E$, and $F$ are constants.

Step 2: Substitute the Points into the General Form

Substitute the coordinates of the points $A$, $B$, and $C$ into the equation to get three equations.

For point $A(2, 3)$: $(2)^2 + (3)^2 + 2D + 3E + F = 0$ $4 + 9 + 2D + 3E + F = 0 \quad \text{(Equation 1)}$ $2D + 3E + F = -13$

For point $B(6, 1)$: $(6)^2 + (1)^2 + 6D + 1E + F = 0$ $36 + 1 + 6D + E + F = 0 \quad \text{(Equation 2)}$ $6D + E + F = -37$

For point $C(4, -3)$: $(4)^2 + (-3)^2 + 4D - 3E + F = 0$ $16 + 9 + 4D - 3E + F = 0 \quad \text{(Equation 3)}$ $4D - 3E + F = -25$

Step 3: Solve the System of Equations

We now have the following system of linear equations:

2D + 3E + F &= -13 \quad \text{(1)}\\ 6D + E + F &= -37 \quad \text{(2)}\\ 4D - 3E + F &= -25 \quad \text{(3)} \end{align*}$$ Subtract Equation 1 from Equation 2: $$(6D + E + F) - (2D + 3E + F) = -37 - (-13)$$ $$4D - 2E = -24$$ $$2D - E = -12 \quad \text{(4)}$$ Subtract Equation 3 from Equation 2: $$(6D + E + F) - (4D - 3E + F) = -37 - (-25)$$ $$2D + 4E = -12 \quad \text{(5)}$$ ### Step 4: Solve for $$D$$, $$E$$, and $$F$$ From Equation 4: $$E = 2D + 12$$ Substitute into Equation 5: $$2D + 4(2D + 12) = -12$$ $$2D + 8D + 48 = -12$$ $$10D = -60$$ $$D = -6$$ Now, substitute $$D = -6$$ into $$E = 2D + 12$$: $$E = 2(-6) + 12 = 0$$ Substitute $$D = -6$$ and $$E = 0$$ into Equation 1 to find $$F$$: $$2(-6) + 3(0) + F = -13$$ $$-12 + F = -13$$ $$F = -1$$ ### Step 5: Write the Equation Substituting $$D = -6$$, $$E = 0$$, and $$F = -1$$ into the general form: $$x^2 + y^2 - 6x - 1 = 0$$ This is the general form of the circle passing through the points $$A(2, 3)$$, $$B(6, 1)$$, and $$C(4, -3)$$. Would you like to see more details or have any questions? Here are 5 related questions: 1. How do you convert the general form of a circle into its standard form? 2. How can you find the center and radius of a circle from the general form? 3. What is the significance of the coefficients $$D$$, $$E$$, and $$F$$ in the general form of a circle? 4. How do you find the equation of a circle given its center and radius? 5. How do you determine if three points lie on a circle? **Tip:** When solving a system of equations, it's often helpful to eliminate variables step by step to simplify the problem.