Question

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

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

Answer 1

The correct option is D None of these.

Since $PQ$ is the chord of contact of the tangents from the origin $O$ to the circle
$x^2+y^2+2gx+2fy+c=0,\left(1\right)$

Equation of $PQ$ is $gx+fy+c=0\left(2\right)$

An equation of a circle through the intersection of (1) and (2) is given by
$x^2+y^2+2gx+2fy+c+\lambda (gx+fy+c)=0\left(3\right)$
If,
the circle $\left(3\right)$ passes through origin, then $c+\lambda c=0,$ i.e $\lambda =-1$
and the equation of the circle $\left(3\right)$
becomes $x^2+y^2+gx+fy=0$

Centre of this circle is $(-g/2,-f/2),$ and hence it is the circumcentre of the triangle $OPQ$