Another approach to the problem is described here, here and here (and the MATLAB source code).

An efficient method of locating these solutions exploits the homogeneous matrix representation of conic sections, i.e. a 3x3 symmetric matrix which depends on six parameters.

Thank you for solving. That is one big formula.

You only have one equation. Eliminating one variable from one equation leaves nothing (a vacuous system, hence satisfied for all values). Instead do the elimination on those two quadrics, each set to zero.

Eliminate[{A1*x*x + B1*x*y + C1*y*y + D1*x + S1*y + F1 == 
   0, A2*x*x + B2*x*y + C2*y*y + D2*x + S2*y + F2 == 0}, x]

(* Out[1794]= 
A2^2 F1^2 + 
  F1 (-A2 D1 D2 + A1 D2^2 - 2 A1 A2 F2 - A2 B2 D1 y - A2 B1 D2 y + 
     2 A1 B2 D2 y + 2 A2^2 S1 y - 2 A1 A2 S2 y - A2 B1 B2 y^2 + 
     A1 B2^2 y^2 + 2 A2^2 C1 y^2 - 2 A1 A2 C2 y^2) == -A2 D1^2 F2 + 
  A1 D1 D2 F2 - A1^2 F2^2 - 2 A2 B1 D1 F2 y + A1 B2 D1 F2 y + 
  A1 B1 D2 F2 y + A2 D1 D2 S1 y - A1 D2^2 S1 y + 2 A1 A2 F2 S1 y - 
  A2 D1^2 S2 y + A1 D1 D2 S2 y - 2 A1^2 F2 S2 y - A2 C2 D1^2 y^2 + 
  A2 C1 D1 D2 y^2 + A1 C2 D1 D2 y^2 - A1 C1 D2^2 y^2 - 
  A2 B1^2 F2 y^2 + A1 B1 B2 F2 y^2 + 2 A1 A2 C1 F2 y^2 - 
  2 A1^2 C2 F2 y^2 + A2 B2 D1 S1 y^2 + A2 B1 D2 S1 y^2 - 
  2 A1 B2 D2 S1 y^2 - A2^2 S1^2 y^2 - 2 A2 B1 D1 S2 y^2 + 
  A1 B2 D1 S2 y^2 + A1 B1 D2 S2 y^2 + 2 A1 A2 S1 S2 y^2 - 
  A1^2 S2^2 y^2 + A2 B2 C1 D1 y^3 - 2 A2 B1 C2 D1 y^3 + 
  A1 B2 C2 D1 y^3 + A2 B1 C1 D2 y^3 - 2 A1 B2 C1 D2 y^3 + 
  A1 B1 C2 D2 y^3 + A2 B1 B2 S1 y^3 - A1 B2^2 S1 y^3 - 
  2 A2^2 C1 S1 y^3 + 2 A1 A2 C2 S1 y^3 - A2 B1^2 S2 y^3 + 
  A1 B1 B2 S2 y^3 + 2 A1 A2 C1 S2 y^3 - 2 A1^2 C2 S2 y^3 + 
  A2 B1 B2 C1 y^4 - A1 B2^2 C1 y^4 - A2^2 C1^2 y^4 - A2 B1^2 C2 y^4 + 
  A1 B1 B2 C2 y^4 + 2 A1 A2 C1 C2 y^4 - A1^2 C2^2 y^4 *)

You use = in three places where you should use == and those mean completely different things to Mathematica. Those might just be typos or they might be a deeper misunderstanding. If those were not just typos then look up = and == in the help pages and study those until you are certain what each means and how they are different.

A completely different issue is, although the documentation for Eliminate doesn't say this, if you are eliminating n variables then it seems that you almost always need at least n+1 equations to get useful and interesting results. Trying to eliminate one variable given only one equation doesn't seem to usually work well. It is perhaps unfortunate that there isn't a warning of some kind to let a user know this.