Find the parametric representation of circles :
(i) `x^(2) + y^(2) + 12x - 4y - 1 = 0`
(ii) `x^(2) + y^(2) + 2gx + 2fy + c = 0 `.

Answer

To find the parametric representation of the given circles, we will follow a systematic approach for both parts of the question.

### Part (i): Given Circle Equation
**Equation:**
\[ x^2 + y^2 + 12x - 4y - 1 = 0 \]

**Step 1: Rearranging the Equation**
We need to rearrange the equation to the standard form of a circle, which is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]

**Step 2: Completing the Square for x and y**
1. For \(x\):
- The terms involving \(x\) are \(x^2 + 12x\).
- To complete the square:
\[ x^2 + 12x = (x + 6)^2 - 36 \]

2. For \(y\):
- The terms involving \(y\) are \(y^2 - 4y\).
- To complete the square:
\[ y^2 - 4y = (y - 2)^2 - 4 \]

**Step 3: Substitute Back into the Equation**
Substituting the completed squares back into the equation:
\[ (x + 6)^2 - 36 + (y - 2)^2 - 4 - 1 = 0 \]
This simplifies to:
\[ (x + 6)^2 + (y - 2)^2 - 41 = 0 \]
Thus:
\[ (x + 6)^2 + (y - 2)^2 = 41 \]

**Step 4: Identify the Center and Radius**
From the equation \((x + 6)^2 + (y - 2)^2 = 41\):
- Center \((h, k) = (-6, 2)\)
- Radius \(r = \sqrt{41}\)

**Step 5: Write the Parametric Equations**
The parametric equations for the circle are:
\[ x = h + r \cos \theta = -6 + \sqrt{41} \cos \theta \]
\[ y = k + r \sin \theta = 2 + \sqrt{41} \sin \theta \]

### Part (ii): General Circle Equation
**Equation:**
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

**Step 1: Rearranging the Equation**
Rearranging to the standard form:
\[ x^2 + 2gx + y^2 + 2fy + c = 0 \]

**Step 2: Completing the Square**
1. For \(x\):
- \(x^2 + 2gx = (x + g)^2 - g^2\)

2. For \(y\):
- \(y^2 + 2fy = (y + f)^2 - f^2\)

**Step 3: Substitute Back into the Equation**
Substituting back gives:
\[ (x + g)^2 - g^2 + (y + f)^2 - f^2 + c = 0 \]
This simplifies to:
\[ (x + g)^2 + (y + f)^2 = g^2 + f^2 - c \]

**Step 4: Identify the Center and Radius**
From the equation \((x + g)^2 + (y + f)^2 = g^2 + f^2 - c\):
- Center \((h, k) = (-g, -f)\)
- Radius \(r = \sqrt{g^2 + f^2 - c}\)

**Step 5: Write the Parametric Equations**
The parametric equations for the circle are:
\[ x = -g + \sqrt{g^2 + f^2 - c} \cos \theta \]
\[ y = -f + \sqrt{g^2 + f^2 - c} \sin \theta \]

### Summary of Solutions
1. For the first circle:
- \(x = -6 + \sqrt{41} \cos \theta\)
- \(y = 2 + \sqrt{41} \sin \theta\)

2. For the second circle:
- \(x = -g + \sqrt{g^2 + f^2 - c} \cos \theta\)
- \(y = -f + \sqrt{g^2 + f^2 - c} \sin \theta\)