Can anyone help me with fitting an implicit function to data?

The function is given by:

    a*x^2 + 2*b*x*y + c*y^2 = e

under the condition

    a*c-b^2=1

something like a least square fit with $\chi^2$ value would be really nice.
Unfortunately I have no idea how to do a fit with an only implicitly given function.

I tried the solutions according to this thread http://mathematica.stackexchange.com/questions/10089/how-to-use-findfit-to-fit-an-implicit-function, but i got the following errors:

    pts = {{0.943851, 0.0255852}, {-2.23615, 0.0255852}, {2.12656, -0.0845208}, {2.32489, -0.0354676}, {-0.284106, -0.0354676}, {-2.64798, 0.0415494},{1.41402,0.0415494}, {-0.114051, 0.0669646}, {-2.99405,0.0669646}, {-3.92389,0.0825293}, {-0.273894,0.0825293}, {-2.66303, 0.0877706}, {-1.76303,0.0877706}}

    fitfunc[a_, b_, d_, x_] := y /. Solve[a*x^2 + 2*b*x*y + (1 + b^2)/a*y^2 == d, {y},InverseFunctions -> True]
    FindFit[pts, fitfunc[a, b, d, x], {a, b, d}, x]
    FindFit::eqineq: Constraints in {(-a b x+Sqrt[a d+a b^2 d-a^2 x^2])/(1+b^2)} are not all equality or inequality constraints. With the exception of integer domain constraints for linear programming, domain constraints or constraints with Unequal (!=) are not supported. >>

I also tried:

    fitfunc2[a_?NumericQ, b_?NumericQ, d_?NumericQ, , x_?NumericQ] :=  y /. FindRoot[a*x^2 + 2*b*x*y + (1 + b^2)/a*y^2 == d, {y, 1.}]
    FindFit[pts, fitfunc2[a, b, d, x], {a, b, d}, x]
    FindFit::nrlnum: The function value {-0.0255852+fitfunc2[1.,1.,1.,0.943851],-0.0255852+fitfunc2[1.,1.,1.,-2.23615],0.0845208\[VeryThinSpace]+fitfunc2[1.,1.,1.,2.12656],0.0354676\[VeryThinSpace]+fitfunc2[1.,1.,1.,2.32489],0.0354676\[VeryThinSpace]+fitfunc2[1.,1.,1.,-0.284106],-0.0415494+fitfunc2[1.,1.,1.,-2.64798],-0.0415494+fitfunc2[1.,1.,1.,1.41402],-0.0669646+fitfunc2[1.,1.,1.,-0.114051],-0.0669646+fitfunc2[1.,1.,1.,-2.99405],-0.0825293+fitfunc2[1.,1.,1.,-3.92389],-0.0825293+fitfunc2[1.,1.,1.,-0.273894],-0.0877706+fitfunc2[1.,1.,1.,-2.66303],-0.0877706+fitfunc2[1.,1.,1.,-1.76303]}
    is not a list of real numbers with dimensions {13} at {a,b,d} = {1.,1.,1.}. >>