If the equation of incircle of an equilateral triangle is `x^(2)+y^(2) + 4x -6y + 4 =0`, then the equation of circumcircle of the triangle is

Options

A. `(-2, -3), 6`
B. `(-2, 3), 6`
C. `(2, 3), 6`
D. none

Answer

To find the equation of the circumcircle of an equilateral triangle given the equation of its incircle, we can follow these steps:

### Step 1: Rewrite the equation of the incircle
The given equation of the incircle is:
\[ x^2 + y^2 + 4x - 6y + 4 = 0 \]

### Step 2: Complete the square
We will complete the square for both \(x\) and \(y\).

For \(x\):
\[ x^2 + 4x = (x + 2)^2 - 4 \]

For \(y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]

Substituting these into the equation:
\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 + 4 = 0 \]
This simplifies to:
\[ (x + 2)^2 + (y - 3)^2 - 9 = 0 \]
or
\[ (x + 2)^2 + (y - 3)^2 = 3^2 \]

### Step 3: Identify the center and radius of the incircle
From the equation \((x + 2)^2 + (y - 3)^2 = 3^2\), we can see that:
- The center of the incircle (and also the circumcircle) is at \((-2, 3)\).
- The radius of the incircle is \(3\).

### Step 4: Find the radius of the circumcircle
For an equilateral triangle, the radius of the circumcircle (denoted as \(R\)) is related to the radius of the incircle (denoted as \(r\)) by the formula:
\[ R = \frac{2}{\sqrt{3}} \times r \]
Substituting \(r = 3\):
\[ R = \frac{2}{\sqrt{3}} \times 3 = 2\sqrt{3} \]

### 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 \(h = -2\), \(k = 3\), and \(R = 2\sqrt{3}\):
\[ (x + 2)^2 + (y - 3)^2 = (2\sqrt{3})^2 \]
This simplifies to:
\[ (x + 2)^2 + (y - 3)^2 = 12 \]

### Step 6: Expand the equation
Expanding the equation:
\[ (x + 2)^2 + (y - 3)^2 = 12 \]
gives:
\[ x^2 + 4x + 4 + y^2 - 6y + 9 = 12 \]
Combining like terms:
\[ x^2 + y^2 + 4x - 6y + 13 - 12 = 0 \]
This simplifies to:
\[ x^2 + y^2 + 4x - 6y + 1 = 0 \]

### Final Answer
The equation of the circumcircle of the triangle is:
\[ x^2 + y^2 + 4x - 6y + 1 = 0 \]

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