{x1,y1}=RandomReal[{-5,5},2]
{x2,y2}=RandomReal[{-5,5},2]
{x3,y3}=RandomReal[{-5,5},2]
{x4,y4}=RandomReal[{-5,5},2]
{x5,y5}=RandomReal[{-5,5},2]

pts={{x1,y1},{x2,y2},{x3,y3},{x4,y4},{x5,y5}}

eq=a x^2+b x y + c y^2+d x + e y +1

{
eq==0/.{x->x1,y->y1},
eq==0/.{x->x2,y->y2},
eq==0/.{x->x3,y->y3},
eq==0/.{x->x4,y->y4},
eq==0/.{x->x5,y->y5}
}

sol=Solve[%]

eqf=eq/.sol[[1]]

cpl=ContourPlot[eqf==0,{x,-6,6},{y,-6,6}]

ptss=Graphics[{AbsolutePointSize[7],Red,Point[pts]}]

Show[cpl,ptss]

ContourPlot[eqf==0,{x,-6,6},{y,-6,6},Epilog->{AbsolutePointSize[7],Red,Point[pts]}]