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 given its centroid and one vertex, we will follow these steps:

### Step 1: Identify the given points
The centroid \( G \) of the triangle is given as \( (2, -2) \) and one vertex \( A \) is given as \( (-1, 1) \).

### Step 2: Find the coordinates of the other two vertices
In an equilateral triangle, the centroid divides each median in the ratio 2:1. Since we know one vertex and the centroid, we can find the other two vertices using the properties of equilateral triangles.

Let the other two vertices be \( B(x_1, y_1) \) and \( C(x_2, y_2) \). The centroid \( G \) can be expressed as:
\[ G = \left(\frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}\right) \]
Substituting the known values:
\[ (2, -2) = \left(\frac{-1 + x_1 + x_2}{3}, \frac{1 + y_1 + y_2}{3}\right) \]
From this, we can derive two equations:
1. \( -1 + x_1 + x_2 = 6 \) (Multiplying both sides by 3)
2. \( 1 + y_1 + y_2 = -6 \)

From the first equation:
\[ x_1 + x_2 = 7 \quad \text{(Equation 1)} \]
From the second equation:
\[ y_1 + y_2 = -7 \quad \text{(Equation 2)} \]

### Step 3: Use the properties of the equilateral triangle
The distance between any two vertices of an equilateral triangle is equal. We can use the distance formula to set up equations based on the distances \( AB \), \( AC \), and \( BC \).

Using the distance formula:
\[ AB = \sqrt{(x_1 + 1)^2 + (y_1 - 1)^2} \]
\[ AC = \sqrt{(x_2 + 1)^2 + (y_2 - 1)^2} \]
\[ BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Since \( AB = AC = BC \), we can set up equations based on these distances.

### Step 4: Find the circumradius
The circumradius \( R \) of an equilateral triangle can be calculated using the formula:
\[ R = \frac{a}{\sqrt{3}} \]
where \( a \) is the length of a side. We can find \( a \) using the distance between the vertex \( A \) and the centroid \( G \):
\[ AG = \sqrt{(2 - (-1))^2 + (-2 - 1)^2} = \sqrt{(3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} \]
Thus, the radius \( R \) is:
\[ R = \frac{\sqrt{18}}{\sqrt{3}} = \sqrt{6} \]

### Step 5: Write the equation of the circumcircle
The equation of a circle with center \( (h, k) \) and radius \( r \) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = 2 \), \( k = -2 \), and \( r = \sqrt{6} \):
\[ (x - 2)^2 + (y + 2)^2 = 6 \]

### Step 6: Expand the equation
Expanding the equation:
\[ (x - 2)^2 + (y + 2)^2 = 6 \]
\[ x^2 - 4x + 4 + y^2 + 4y + 4 = 6 \]
\[ x^2 + y^2 - 4x + 4y + 8 - 6 = 0 \]
\[ x^2 + y^2 - 4x + 4y + 2 = 0 \]

### Final Equation
The equation of the circumcircle is:
\[ x^2 + y^2 - 4x + 4y + 2 = 0 \]