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(D) The equation of the pair of tangents from a point $(x_1, y_1)$ to a circle $S = 0$ is given by $SS_1 = T^2$.Here,the point is the origin $(0, 0)$,

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$(gx + fy)^2 = c(x^2 + y^2)$

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(D) The equation of the pair of tangents from a point $(x_1, y_1)$ to a circle $S = 0$ is given by $SS_1 = T^2$.Here,the point is the origin $(0, 0)$,so $x_1 = 0$ and $y_1 = 0$.$S = x^2 + y^2 + 2gx + 2fy + c = 0$$S_1 = 0^2 + 0^2 + 2g(0) + 2f(0) + c = c$$T = x(0) + y(0) + g(x + 0) + f(y + 0) + c = gx + fy + c$Substituting these into $SS_1 = T^2$:$c(x^2 + y^2 + 2gx + 2fy + c) = (gx + fy + c)^2$$c(x^2 + y^2) + 2gcx + 2fcy + c^2 = (gx + fy)^2 + 2c(gx + fy) + c^2$$c(x^2 + y^2) + 2gcx + 2fcy + c^2 = (gx + fy)^2 + 2gcx + 2fcy + c^2$$c(x^2 + y^2) = (gx + fy)^2$.

The equation of the pair of tangents drawn from the origin to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is:

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