STATEMENT -1 : if O is the origin and OP and OQ are tangents to the circle `x^(2) + y^(2) + 2x + 4y + 1 = 0` the circumcentre of the triangle is `((-1)/(2), -1)` .
and
STATEMENT-2 : `OP.OQ = PQ^(2)`.

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 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

Answer

To solve the problem, we will analyze the statements given and verify their correctness step by step.

### Step 1: Rewrite the Circle's Equation
The given equation of the circle is:
\[ x^2 + y^2 + 2x + 4y + 1 = 0 \]
We can rewrite this in standard form by completing the square.

### Step 2: Complete the Square
1. For \(x^2 + 2x\):
\[ x^2 + 2x = (x + 1)^2 - 1 \]
2. For \(y^2 + 4y\):
\[ y^2 + 4y = (y + 2)^2 - 4 \]

Substituting back into the equation:
\[ (x + 1)^2 - 1 + (y + 2)^2 - 4 + 1 = 0 \]
This simplifies to:
\[ (x + 1)^2 + (y + 2)^2 - 4 = 0 \]
Thus, we have:
\[ (x + 1)^2 + (y + 2)^2 = 4 \]
This shows that the circle has a center at \((-1, -2)\) and a radius of \(2\).

### Step 3: Verify the Origin's Position
To check if the origin \((0, 0)\) lies outside the circle, we substitute \((0, 0)\) into the circle's equation:
\[ (0 + 1)^2 + (0 + 2)^2 = 1 + 4 = 5 \]
Since \(5 > 4\), the origin is indeed outside the circle.

### Step 4: Determine the Tangents from the Origin
The tangents from the origin to the circle can be found using the formula:
\[ OP \cdot OQ = r^2 \]
Where \(r\) is the radius of the circle. Here, \(r = 2\), so:
\[ OP \cdot OQ = 2^2 = 4 \]

### Step 5: Find the Circumcenter of Triangle OPQ
The circumcenter of triangle \(OPQ\) (where \(OP\) and \(OQ\) are tangents) lies on the line joining the center of the circle and the origin. The circumcenter can be calculated as the midpoint of the line segment joining the center of the circle and the origin:
\[ \text{Circumcenter} = \left( \frac{0 + (-1)}{2}, \frac{0 + (-2)}{2} \right) = \left( -\frac{1}{2}, -1 \right) \]
Thus, Statement 1 is true.

### Step 6: Verify Statement 2
We need to check if:
\[ OP \cdot OQ = PQ^2 \]
From the previous calculations, we found \(OP \cdot OQ = 4\).

To find \(PQ\), we can use the fact that \(P\) and \(Q\) are points where the tangents touch the circle. The length \(PQ\) can be calculated using the properties of tangents:
\[ PQ = 2 \cdot OP \cdot \sin(\theta) \]
Where \(\theta\) is the angle between the radius and the tangent. However, since \(OP\) and \(OQ\) are equal, we can simplify:
\[ PQ^2 = (2 \cdot OP)^2 = 4 \cdot OP^2 \]
Since \(OP^2\) does not equal \(4\) (as \(OP\) is not equal to \(2\)), thus:
\[ OP \cdot OQ \neq PQ^2 \]
Therefore, Statement 2 is false.

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

### Final Answer
The correct option is **C** (Statement 1 is true, Statement 2 is false).