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 ($\pm\delta a$, $\pm\delta b$, $\pm\delta c$, $\pm\delta d$). 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 $A=\begin{pmatrix}a^2 & b^2 \\ c^2 & d^2\end{pmatrix}$, $X=\begin{pmatrix}x^2 \\ y^2\end{pmatrix}$, $Y=\begin{pmatrix}1 \\ 1\end{pmatrix}$.
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 $(\hat{x},\hat{y})$.
Let $a,b,c,d$ be the "correct" parameters and $a + \delta a$, ..., $ d + \delta d$ the measured parameters including errors. The resulting matrix is $A + \delta A$ where to first order $$ \delta A \approx \pmatrix{2 a\; \delta a & 2 b \; \delta b\cr 2 c\; \delta c & 2 d\; \delta d}$$ and you compute $X + \delta X = (A + \delta A)^{-1} Y$. Again to first order, assuming $A$ is invertible, $$(A + \delta A)^{-1} \approx A^{-1} + A^{-1} (\delta A) A^{-1}$$ so $$\delta X \approx A^{-1} (\delta A) A^{-1} Y $$