Statement-1: The equation `x^(2)-y^(2)-4x-4y=0` represents a circle with centre (2, 2) passing through the origin.
Statement-2: The equation `x^(2)+y^(2)+4x+6y+13=0` represents a point.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Explanation

To solve the given statements, we will analyze each statement step by step.

### Statement 1:
The equation \( x^2 - y^2 - 4x - 4y = 0 \) represents a circle with center (2, 2) passing through the origin.

**Step 1: Rearranging the equation**
We start with the equation:
\[ x^2 - y^2 - 4x - 4y = 0 \]
Rearranging gives:
\[ x^2 - 4x - y^2 - 4y = 0 \]

**Step 2: Identifying the form of the equation**
The general equation of a circle is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius. However, our equation has a minus sign between the \(x^2\) and \(y^2\) terms, indicating that it is not in the form of a circle.

**Step 3: Conclusion for Statement 1**
Since the equation has a minus sign between \(x^2\) and \(y^2\), it does not represent a circle. Therefore, Statement 1 is **false**.

### Statement 2:
The equation \( x^2 + y^2 + 4x + 6y + 13 = 0 \) represents a point.

**Step 1: Rearranging the equation**
We start with the equation:
\[ x^2 + y^2 + 4x + 6y + 13 = 0 \]
We can rearrange it to group the \(x\) and \(y\) terms:
\[ x^2 + 4x + y^2 + 6y + 13 = 0 \]

**Step 2: Completing the square for \(x\)**
For the \(x\) terms:
\[ x^2 + 4x = (x + 2)^2 - 4 \]

**Step 3: Completing the square for \(y\)**
For the \(y\) terms:
\[ y^2 + 6y = (y + 3)^2 - 9 \]

**Step 4: Substitute back into the equation**
Substituting these back into the equation gives:
\[ ((x + 2)^2 - 4) + ((y + 3)^2 - 9) + 13 = 0 \]
This simplifies to:
\[ (x + 2)^2 + (y + 3)^2 - 4 - 9 + 13 = 0 \]
\[ (x + 2)^2 + (y + 3)^2 = 0 \]

**Step 5: Conclusion for Statement 2**
The equation \((x + 2)^2 + (y + 3)^2 = 0\) implies that both squares must equal zero, which means:
\[ x + 2 = 0 \quad \text{and} \quad y + 3 = 0 \]
Thus, \(x = -2\) and \(y = -3\). This represents a single point \((-2, -3)\). Therefore, Statement 2 is **true**.

### Final Conclusion:
- Statement 1 is **false**.
- Statement 2 is **true**.