Consider a regular surface $M=(a(x_1,x_2),b(x_1,x_2),c(x_1,x_2)) $. The matrix of the I Fundamental Form is given by $g_{i,j}= \langle M_i, M_j\rangle $, where $M_i$ is the tangent vector $\frac{\partial M}{\partial x_i}$. The matrix coefficients for the II Fundamental Form are given by $h_{ij}=-\langle N, x_{ij}\rangle$, here $N$ is the normal vector. Consider the diagonal matrix with $\lambda_1,\lambda_2,\lambda_3$ in the diagonal.
I have the following expression
$$[M_1,M_2]^T[\lambda][M_1,M_2] - \gamma h_{i,j}$$
and I would like to know for each $\gamma$ and $\lambda$'s the above matrix has a zero determinant, or equivalent when its rank in less than 2.
I noticed that $[M_1, M_2]^T [M_1,M_2]$ is precisely the matrix of the I Fundamental Form, but this didn't help me a lot.