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
$x^2+y^2+2gx+2fy+c=0,$---------------(1)
equation of PQ is
$gx+fy+c=0$---------------------------------- (2)
An equation of a circle through the intersection of (1) and (2) is given by
$x^2+y^2+2gx+2fy+c+\lambda (gx+fy+c)=0$ -----------(3)
If the circle (3) passes through O, the origin, then $c+\lambda c=0,$ i.e., $\lambda =-1,$ and the equation of the circle (3) becomes
$x^2+y^2+gx+fy=0$
Centre of this circle is (-g/2, -f/2), and hence it is the circum centre of the triangle OPQ.