To find the equation of the circle with center (-1, 2) that passes through the centroid of the triangle formed by the points (3, 1), (2, -1), and (1, 3), we will follow these steps:
### Step 1: Find the Centroid of the Triangle
The formula for the centroid (C) of a triangle formed by the vertices (x1, y1), (x2, y2), and (x3, y3) is given by:
\[
C = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\]
For our triangle, the vertices are (3, 1), (2, -1), and (1, 3).
Calculating the x-coordinate of the centroid:
\[
x_C = \frac{3 + 2 + 1}{3} = \frac{6}{3} = 2
\]
Calculating the y-coordinate of the centroid:
\[
y_C = \frac{1 + (-1) + 3}{3} = \frac{3}{3} = 1
\]
Thus, the coordinates of the centroid are \( C(2, 1) \).
### Step 2: Calculate the Radius of the Circle
The radius (r) of the circle is the distance from the center (-1, 2) to the centroid (2, 1). We use the distance formula:
\[
r = \sqrt{(x_C - x_O)^2 + (y_C - y_O)^2}
\]
Substituting the coordinates:
\[
r = \sqrt{(2 - (-1))^2 + (1 - 2)^2} = \sqrt{(2 + 1)^2 + (1 - 2)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]
### Step 3: Write the Equation of the Circle
The standard equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Here, the center is \((-1, 2)\) and the radius is \(\sqrt{10}\). Therefore, the equation becomes:
\[
(x - (-1))^2 + (y - 2)^2 = (\sqrt{10})^2
\]
This simplifies to:
\[
(x + 1)^2 + (y - 2)^2 = 10
\]
### Step 4: Expand the Equation
Expanding the left side:
\[
(x + 1)^2 = x^2 + 2x + 1
\]
\[
(y - 2)^2 = y^2 - 4y + 4
\]
Combining these:
\[
x^2 + 2x + 1 + y^2 - 4y + 4 = 10
\]
This simplifies to:
\[
x^2 + y^2 + 2x - 4y + 5 = 10
\]
Rearranging gives:
\[
x^2 + y^2 + 2x - 4y - 5 = 0
\]
### Final Answer
The equation of the circle is:
\[
x^2 + y^2 + 2x - 4y - 5 = 0
\]
