Intersection of 4D curves, one is 3-sphere other is the determinant of the four dimensions

Question

Let the curve with 3-sphere be represented by equation, $$a_{11}^2 + a_{12}^2 + a_{21}^2+ a_{22}^2 = C_1$$

And the curve with the determinant of matrix, $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ is,

$$a_{11}a_{22} - a_{12}a_{21} = C_2 $$

Is there a unique intersection that can be determined given this type of curves ? How can I numerically find the intersection between the two hypersurfaces, given two positive real constants $C_1>0$ and $C_2>0$ ?

Edit: 11.07pm, 9/20/2015 -- I am using MATLAB but the command script seems to be incorrect, I don't know how to interpret it, consider,

syms a b c d C_1 C_2
assume(a, 'integer')
assume(b, 'integer')
assume(d, 'integer')
assume(c, 'integer')
assume(a > 0)
assume(b > 0)
assume(c > 0)
assume(d > 0)
assume(C_1 > 0)
assume(C_2 > 0)
assume(C_1, 'real')
assume(C_2, 'real')
eqn1 = a^2+b^2+c^2+d^2 == C_1;
eqn2 =  a*d - c*d == C_2;
[S] = solve ( [eqn1,eqn2], [a,b,c,d], 'ReturnConditions', true );
S.a
S.b
S.c
S.d

I get the answer,

S.parameters
ans =
[ z, z1]
ans =
             z1
 (C_2 + z1*z)/z
 (C_2 + z1*z)/z

ans =
                                 (- z^2 - 2*z1^2 + C_1)^(1/2)
  (- C_2^2 - 2*C_2*z*z1 - z^4 - 2*z^2*z1^2 + C_1*z^2)^(1/2)/z
 -(- C_2^2 - 2*C_2*z*z1 - z^4 - 2*z^2*z1^2 + C_1*z^2)^(1/2)/z

ans =
 z1
 z1
 z1

ans =
 z
 z
 z