Equation of Parabola through Four Points

{x1,y1,x2,y2,x3,y3}={-2.,3.,1.,-1.,4.,0};
soln=NSolve[{h^2==a b,a x^2+2 h x y+b y^2+2 f x+2 g y==1,a x1^2+2 h x1 y1+b y1^2+2 f x1+2 g y1==1,a x2^2+2 h x2 y2+b y2^2+2 f x2+2 g y2==1,a x3^2+2 h x3 y3+b y3^2+2 f x3+2 g y3==1},{a,b,h,f,g}]

Attempting to find constant coefficients from standard conic equation through three fixed points [(x1,y1),(x2,y2),(x3,y3)] and a variable point $(x,y)$ along with a condition for parabola. Errors in realization of the second degree result. Appreciate all help.

EDIT1: 

This worked earlier but does not, anymore.

   {x1, y1, x2, y2, x3, y3} = {-2., 3, 1., -1, 4, 0};
equn = {h^2 == a b,
   a x^2 + 2 h x y + b y^2 + 2 f x + 2 g y == 1,
   a x1^2 + 2 h x1 y1 + b y1^2 + 2 f x1 + 2 g y1 == 1,
   a x2^2 + 2 h x2 y2 + b y2^2 + 2 f x2 + 2 g y2 == 1,
   a x3^2 + 2 h x3 y3 + b y3^2 + 2 f x3 + 2 g y3 == 1};
sol = NSolve[equn, {a, b, h, f, g}];