I have the following 4 equations:
\begin{eqnarray} z &=& c_0 + c_1x + c_2y + c_3xy + c_4x^2 + c_5y^2 \\ 2(x-x_0) &=& \lambda(c_1 + c_3y + 2c_4x) \\ 2(y-y_0) &=& \lambda(c_2 + c_3x + 2c_5y) \\ 2(z-z_0) &=& -\lambda \end{eqnarray}
where $x, y, z$, and $\lambda$ are unknown, $c_i$ are known coefficients, and $x_0, y_0$, and $z_0$ are also known. Am I right in that this is a non-linear system of equations? Is there a way that I can solve this analytically? I'd prefer to not employ a numerical approach, if possible.
Choosing a convenient change of coordinates like
$$ \cases{ x = \cos\theta X+\sin\theta Y\\ y = -\sin\theta x+\cos\theta Y\\ z = Z } $$
with a convenient $\theta$ we can reduce the system to
$$ \cases{Z=a_0+a_1 X+ a_2 Y + a_3 X^2+ a_4 Y^2\\ 2(X-X_0)=\lambda(a_1+2a_3 X)\\ 2(Y-Y_0)=\lambda(a_2+2a_4Y)\\ 2(Z-Z_0) = -\lambda } $$
but even in this case, without previous knowledge of the involved parameters, the result is very lengthy. (As can be verified with the help of a symbolic processor like MATHEMATICA)