To find the equation of the circle passing through three points, we use the general equation of a circle:

$x^2 + y^2 + Dx + Ey + F = 0$

Where $D$, $E$, and $F$ are constants to be determined. We can substitute the coordinates of the given points $(5,7)$, $(8,1)$, and $(1,3)$ into the equation to form a system of equations.

Step 1: Substitute each point into the general equation

1. For $(5,7)$: $5^2 + 7^2 + 5D + 7E + F = 0$ $25 + 49 + 5D + 7E + F = 0 \quad \Rightarrow \quad 5D + 7E + F = -74 \quad \text{(Equation 1)}$

2. For $(8,1)$: $8^2 + 1^2 + 8D + 1E + F = 0$ $64 + 1 + 8D + 1E + F = 0 \quad \Rightarrow \quad 8D + E + F = -65 \quad \text{(Equation 2)}$

3. For $(1,3)$: $1^2 + 3^2 + 1D + 3E + F = 0$ $1 + 9 + D + 3E + F = 0 \quad \Rightarrow \quad D + 3E + F = -10 \quad \text{(Equation 3)}$

Step 2: Solve the system of equations

We now have the following system of linear equations:

5D + 7E + F &= -74 \quad \text{(Equation 1)} \\ 8D + E + F &= -65 \quad \text{(Equation 2)} \\ D + 3E + F &= -10 \quad \text{(Equation 3)} \end{aligned}$$ We can solve this system to find $$D$$, $$E$$, and $$F$$. Let me calculate that.The solution to the system of equations is: $$D = -\frac{29}{3}, \quad E = -\frac{19}{3}, \quad F = \frac{56}{3}$$ ### Step 3: Substitute $$D$$, $$E$$, and $$F$$ into the general equation of the circle The equation of the circle becomes: $$x^2 + y^2 - \frac{29}{3}x - \frac{19}{3}y + \frac{56}{3} = 0$$ Multiplying the entire equation by 3 to eliminate the fractions: $$3x^2 + 3y^2 - 29x - 19y + 56 = 0$$ Thus, the equation of the circle passing through the points $$(5,7)$$, $$(8,1)$$, and $$(1,3)$$ is: $$3x^2 + 3y^2 - 29x - 19y + 56 = 0$$ Would you like more details or have any questions? Here are five related questions to explore: 1. How can we find the center and radius of the circle from its equation? 2. How can we derive the equation of a circle given its center and radius? 3. Can the equation of a circle be represented in a different form, like the standard form? 4. How can we verify if another point lies on this circle? 5. What happens if the points given are collinear? **Tip:** The standard form of a circle's equation is $$(x - h)^2 + (y - k)^2 = r^2$$, where $$(h, k)$$ is the center and $$r$$ is the radius.