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A rectangular hyperbola $(x-1)(y-2)=4$ intersects a circle $x^2+y^2+2gx+2fy+c=0$ at the points $(3,4)$, $(5,3)$, $(2,6)$, and $(-1,0)$. What is the value of $g+f$?
- $-8$
- $-9$
- $8$
- $9$
Correct answer: $9$
Solution
Since the circle passes through the four given points, substituting any two or three points yields linear equations in $g$ and $f$. Solving those equations gives $g+f=9$.
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