Let $C$ be a curve which is locus of point of intersection of the lines $x=2+m$and $my=4-m$. A circle $S:(x-2)^2+(y+1)^2=25 $ intersects the curve $C$ at four points $P,Q,R,S$. If $O$ is the centre of the curve $C$ then find $OP^2+OQ^2+OR^2+OS^2$.
I got the curve $C$ by eliminating $m$ as $$xy-2y+x=6$$ How to find the centre of the curve ? Further how to get the points of intersections ? S0lving the curves simultaneously doesn't seem to be a good idea.
2026-04-19 13:01:51.1776603711
Finding distance of points of intersection of curve with another form its center.
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Rewrite the equation of curve $C$ as follows: $$ (x-2)(y+1)=4. $$ It is then apparent this is the equation of a hyperbola, centered at $O=(2,-1)$. This is also the center of circle $S$, so you don't have to find the coordinates of the intersection points to know their distance from $O$.