If the rectangle hyperbola $(x-1)(y-2)=4$ cuts the circle $x^2+y^2+2gx+2fy+c=0$
at the points $(3,4)\;,(5,3)\;,(2,6)\;,(-1,0)$, then find the value of $g+f$ .
My try: Given that the circle $x^2+y^2+2gx+2fy+c=0$ passes through these $4$ points we have:
$9+16+6g+8f+c=0\Rightarrow 6g+8f+c=-25\cdots (1)$
$25+9+10g+6f+c=0\Rightarrow 10g+6f+c=-34\cdots (2)$
$4+36+4g+12f+c=0\Rightarrow 4g+12f+c=-40\cdots (3)$
$1+0-2g+0+c=0\Rightarrow -2g+c=-1\cdots (4)$
Now we will solve this system of equations for $g$ and $f$ .
But this is very tedious. Could someone explain to me how to solve in an easier way? Thanks.