If O is the origin and OP, OQ are distinct tangents to the circle `x^(2)+y^(2)+2gx+2fy+c=0` then the circumcentre of the triangle OPQ is

Options

A. `(-g,-f)`
B. `(g,f)`
C. `(-f,-g)`
D. None of these

Answer

To solve the problem, we need to find the circumcenter of the triangle formed by the origin \( O \) and the points \( P \) and \( Q \) where \( OP \) and \( OQ \) are distinct tangents to the circle given by the equation:

\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

### Step-by-Step Solution

1. **Identify the Circle's Center and Radius**:
The given circle can be rewritten in standard form. The center of the circle is at \( (-g, -f) \) and the radius \( r \) can be found using the equation:
\[ r = \sqrt{g^2 + f^2 - c} \]

2. **Equation of the Chord of Contact**:
The chord of contact from the origin \( O(0, 0) \) to the circle is given by the equation:
\[ gx + fy + c = 0 \]
This line represents the points \( P \) and \( Q \) where the tangents from the origin touch the circle.

3. **Finding the Circumcircle**:
The circumcircle of triangle \( OPQ \) will pass through the points \( O, P, \) and \( Q \). The circumcircle can be determined using the equations of the circle and the chord of contact.

4. **Forming the Equation of the Circumcircle**:
The equation of the circumcircle can be formed by taking the linear combination of the circle's equation and the chord of contact:
\[ x^2 + y^2 + 2gx + 2fy + c + \lambda(gx + fy + c) = 0 \]
where \( \lambda \) is a parameter.

5. **Setting the Constant Term to Zero**:
Since the circumcircle must pass through the origin \( O(0, 0) \), we set the constant term to zero:
\[ c + \lambda c = 0 \implies c(1 + \lambda) = 0 \]
This gives us two cases: \( c = 0 \) or \( \lambda = -1 \).

6. **Substituting \( \lambda = -1 \)**:
If \( \lambda = -1 \), the equation simplifies to:
\[ x^2 + y^2 + (2g - g)x + (2f - f)y = 0 \]
which simplifies to:
\[ x^2 + y^2 + gx + fy = 0 \]

7. **Finding the Center of the Circumcircle**:
The center of the circumcircle can be found by taking the negative of the coefficients of \( x \) and \( y \) divided by 2:
\[ \text{Center} = \left(-\frac{g}{2}, -\frac{f}{2}\right) \]

### Final Answer
Thus, the circumcenter of triangle \( OPQ \) is:
\[ \left(-\frac{g}{2}, -\frac{f}{2}\right) \]