Tangents drawn from the origin O to the circl | vedclass

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(B) The equation of the chord of contact of tangents drawn from the origin $(0,0)$ to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is given by $T = 0$,w

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$x^2 + y^2 + gx + fy = 0$

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(B) The equation of the chord of contact of tangents drawn from the origin $(0,0)$ to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is given by $T = 0$,which is $gx + fy + c = 0$  $(i)$.The family of circles passing through the intersection of the circle $S = x^2 + y^2 + 2gx + 2fy + c = 0$ and the line $L = gx + fy + c = 0$ is given by $S + \lambda L = 0$.$(x^2 + y^2 + 2gx + 2fy + c) + \lambda(gx + fy + c) = 0$  $(ii)$.Since the circumcircle of $	riangle OPQ$ passes through the origin $(0,0)$,we substitute $x=0, y=0$ into equation $(ii)$:$(0 + 0 + 0 + 0 + c) + \lambda(0 + 0 + c) = 0$$c + \lambda c = 0 \Rightarrow \lambda = -1$ (assuming $c 
eq 0$).Substituting $\lambda = -1$ into equation $(ii)$:$(x^2 + y^2 + 2gx + 2fy + c) - (gx + fy + c) = 0$$x^2 + y^2 + gx + fy = 0$.

Tangents drawn from the origin $O$ to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ touch the circle at the points $P$ and $Q$. Then the equation of the circumcircle of the triangle $OPQ$ is

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