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  4. If O is the origin and OP , OQ are distinct tangents to the circle x2+y2+2gx+2fy+c=0, then the circumcentre of the triangle OPQ is

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Q. 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

NTA AbhyasNTA Abhyas 2020Conic Sections

Solution:

Tangents drawn from the point $O$ , meet the cirle at $P$ & $Q$ and $C$ be the centre of given circle. Then points $O, \, P, \, C$ and $Q$ are concyclic. That means any circle passing through $O, \, P$ and $Q$ also passes through $C$ and $OC$ is the diameter for this circle. Hence mid point of $OC$ is the circumcentre of triangle $OPQ$ .
Solution

Coordinates of circumcenter $=\left(\frac{- g}{2} , \frac{- f}{2}\right)$