question_answer

If O is the origin and OP, OQ  are 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

Correct Answer: D

Solution :

           Equation of pair of tangents from (0, 0) to circle are\[S{{S}_{1}}={{T}^{2}}\].                    Equation of circle through origin and chord of contact is \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c+\lambda (gx+fy+c)=0\]                    \[\Rightarrow \lambda =-1\]    (by\[x=0,\ y=0\])                    Therefore, equation is \[{{x}^{2}}+{{y}^{2}}+gx+fy=0\].                    Hence circumcentre is \[\left( -\frac{g}{2},\ -\frac{f}{2} \right)\]                    Aliter: Required circumcentre is the mid-point of      (0, 0) and \[(-g,\ -f)\] i.e., \[\left( -\frac{g}{2},\ -\frac{f}{2} \right)\].