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Find the condition that the line $lx+my+n=0$ should be a normal to the circle $x^2+y^2+2gx+2fy+c=0$.

My Attempt: Equation of tangent to the circle $x^2+y^2++2gx+2fy+c=0$ at point $P(x_1,y_1)$ is: $$xx_1+yy_1+g(x+x_1)+f(y+y_1)+c=0$$ Slope of tangent$=-\dfrac {x_1+g}{y_1+f}$ Since, tangent is perpendicular to Normal, the slope of normal $=\dfrac {y_1+f}{x_1+g}$

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The line is perpendicular to the circle iff it passes through its center. The circle's equation can be rewritten as $(x+g)^2+(y+f)^2=f^2+g^2-c$, which shows that $(-g, -f)$ is the center.

So the condition becomes $lg+mf=n$

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