Intersection point of two curves with errors - covariance matrix

I have measured the parameters for a hyperbola and an ellipse, let us call them
$$
begin{cases} a^2x^2 + b^2y^2 = 1 \ c^2x^2 + d^2y^2 = 1 end{cases}
$$

and I have errors associated with each parameter a, b, c and d ($pmdelta a$, $pmdelta b$, $pmdelta c$, $pmdelta d$). I want to find the intersection point(s) (due to symmetry there is in practice only one solution), between these two curves, that is, I want to solve the equation:

$$
AX = Y
$$

where I have defined $A=begin{pmatrix}a^2 & b^2 \ c^2 & d^2end{pmatrix}$, $X=begin{pmatrix}x^2 \ y^2end{pmatrix}$, $Y=begin{pmatrix}1 \ 1end{pmatrix}$.

Now, solving the equation is done by simply taking the inverse of A. What I am struggling with however is how to incorporate the errors, such that I derive the covariance matrix for the estimated intersection point $(hat{x},hat{y})$.