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
Step-by-Step Solution
Key Concept: When tangents from an external point touch a circle, the four points (external point, two tangent points, center) form a cyclic quadrilateral with the line joining them as diameter.
Tangents drawn from point $O$ meet the circle at points $P$ and $Q$ with center $C$. Since $O, P, C, Q$ are concyclic (all angles are right angles at $P$ and $Q$), the circle passing through these four points has $OC$ as its diameter. The circumcenter of triangle $OPQ$ is the midpoint of $OC$. Using the property that tangent segments from an external point are equal and the geometry of the configuration determines the answer.
Correct Answer: 4