If the centroid of an equilateral triangle is (2, -2) and its one vertex is (-1, 1) , then the equation of its circumcircle is

Options

A. `x^(2)+y^(2)-4x+4y-10=0`
B. `x^(2)+y^(2)+4x-4y+10=0`
C. `x^(2)+y^(2)+4x-4y-10=0`
D. `x^(2)+y^(2)+4x+4y+10=0`

Explanation

To find the equation of the circumcircle of the equilateral triangle, we can follow these steps:

### Step 1: Understand the properties of the centroid
The centroid (G) of a triangle is the average of the coordinates of its vertices. For an equilateral triangle, the centroid is also the center of the circumcircle.

### Step 2: Use the centroid formula
Let the vertices of the triangle be A(x1, y1), B(x2, y2), and C(x3, y3). The coordinates of the centroid G are given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
Given that G = (2, -2) and one vertex A = (-1, 1), we can set up the equations.

### Step 3: Set up the equations
Let the coordinates of the other two vertices be B(x2, y2) and C(x3, y3). Then, we have:
\[ \frac{-1 + x_2 + x_3}{3} = 2 \quad \text{(1)} \]
\[ \frac{1 + y_2 + y_3}{3} = -2 \quad \text{(2)} \]

### Step 4: Solve equation (1)
From equation (1):
\[ -1 + x_2 + x_3 = 6 \implies x_2 + x_3 = 7 \quad \text{(3)} \]

### Step 5: Solve equation (2)
From equation (2):
\[ 1 + y_2 + y_3 = -6 \implies y_2 + y_3 = -7 \quad \text{(4)} \]

### Step 6: Use the properties of an equilateral triangle
In an equilateral triangle, the distance from the centroid to any vertex is equal to the circumradius (R). The distance from G(2, -2) to A(-1, 1) can be calculated using the distance formula:
\[ R = \sqrt{(2 - (-1))^2 + (-2 - 1)^2} = \sqrt{(2 + 1)^2 + (-2 - 1)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

### Step 7: Write the equation of the circumcircle
The general equation of a circle with center (h, k) and radius R is:
\[ (x - h)^2 + (y - k)^2 = R^2 \]
Substituting h = 2, k = -2, and R = 3√2:
\[ (x - 2)^2 + (y + 2)^2 = (3\sqrt{2})^2 \]
\[ (x - 2)^2 + (y + 2)^2 = 18 \]

### Final Equation
Thus, the equation of the circumcircle is:
\[ (x - 2)^2 + (y + 2)^2 = 18 \]

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