Find the equation of the circle passing through the three points(5,7), (8,1), (1,3)

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:

\[
\begin{aligned}
5D + 7E + F &= -74 \quad \text{(Equation 1)} \\
8D + E + F &= -65 \quad \text{(Equation 2)} \\
D + 3E + F &= -10 \quad \text{(Equation 3)}
\end{aligned}
\]

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.

Find the equation of the circle passing through the three points (5,7), (8,1), (1,3).

Learn how to find the equation of the circle passing through the points (5,7), (8,1), and (1,3). This solution explains key concepts in analytic geometry, using the general equation of a circle and solving a system of equations to determine the constants. Suitable for high school students in Grades 10-12.

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