Mathematica can do this

Solve[{A1*x^2 + B1*x*y + C1*y^2 + D1*x + S1*y + F1 == 0, 
  A2*x^2 + B2*x*y + C2*y^2 + D2*x + S2*y + F2 == 0}, {x, y}]

but the result has 4960729 variables and operators in it and takes a very long time to format and put on the screen.

Simplify might be able to make that somewhat smaller, but would probably take vast amounts of time.

I do not know a way to get WolframAlpha to be able to do this. I tried things like you tried and got the same sqrt x^ 2 response.

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 *)

Using == instead of = does not appear to make a difference in this case.

Here is another approach yet without success:

1 Isolate the 'y' coordinate from the quadratic Hyperbola equation in Wolfram:

Solve[{A1 x^2 + B1 x y + C1 y^2 + D1 x + S1 y + F1 == 0}, {y}]

2 Result from above:

y = (-sqrt((B1 x+S1)^2-4 C1 (A1 x^2+D1 x+F1))-B1 x-S1)/(2 C1) and C1!=0

3 I do the same for the other Hyperbola

y = (-sqrt((B2 x+S2)^2-4 C2 (A2 x^2+D2 x+F2))-B2 x-S2)/(2 C2) and C2!=0

4 I put the two together:

(-sqrt((B1 x+S1)^2-4 C1 (A1 x^2+D1 x+F1))-B1 x-S1)/(2 C1) == (-sqrt((B2 x+S2)^2-4 C2 (A2 x^2+D2 x+F2))-B2 x-S2)/(2 C2)

5 I solve for x in Wolfram:

Solve [{(-sqrt((B1 x+S1)^2-4 C1 (A1 x^2+D1 x+F1))-B1 x-S1)/(2 C1) == (-sqrt((B2 x+S2)^2-4 C2 (A2 x^2+D2 x+F2))-B2 x-S2)/(2 C2)}, {x}]

6 Result from above:

Wolfram|Alpha doesn't understand your query. Showing instead result for query: Solve sqrt x^2