Pont of contact tangent_Circle

Question

The point of contact between a line $lx+my+n$ and the circle $x^2+y^2=a^2$ is $(-a^2l/n,-a^2m/n)$

What is the POC between the same line and the circle $x^2+y^2+2gx+2fy+c$?

Answer 1

Whatever this 'point of contact' means, if we assume that it is invariant under translation by any vector, or in other words, invariant under coordinate transformations $x\to x+g$ and $y\to y+f$, then, introducing the new coordinates of the shifted coordinate system $$X:=x+g,\ Y:=y+f\,,$$ for the circle we have $x^2+y^2+2gx+2fy+c=X^2+Y^2-g^2-f^2+c$, so let us set $A^2:=g^2+f^2-c\ $ (note that it should be positive in order to get a circle), and the line becomes $$lx+my+n=lX+mY-lg-mf+n$$ So, set $N:=n-lg-mf$, and then this 'point of contact' will be $$(X,Y)=\left(-\frac{A^2l}N,-\frac{A^2m}N\right)$$ and so, using $X=x+g$ and $Y=y+f$, we have $$(x,y)=\left(-\frac{A^2l}N-g,-\frac{A^2m}N-f\right)\,.$$