Find locus of points by finding eigenvalues

Some hints (as you requested):

• What we're studying is a second degree equation in $x$ and $y$, so it's a conic section curve (ellipse, hyperbola, or parabola).

• After we find the eigenvalues and eigenvectors, we can use them to diagonalize the matrix $\mathbf{M}$.

• The diagonalization process is really just a change of coordinate system, say from $(x,y)$ coordinates to $(u,v)$ coordinates

• In $(u,v)$ coordinates, since we're now dealing with a diagonal matrix, the given equation takes the form $au^2 + bv^2 = 1$. This is a conic section curve whose geometry you probably understand.

The eigenvectors are orthogonal and span $\Bbb R^2$. This means $\mathbf v=\begin{bmatrix}x\\y\end{bmatrix}=c_1\mathbf x_1+c_2\mathbf x_2$.

$\mathbf v^TM\mathbf v=(c_1\mathbf x_1^T+c_2\mathbf x_2^T)M(c_1\mathbf x_1+c_2\mathbf x_2)\\=(c_1\mathbf x_1^T+c_2\mathbf x_2^T)(c_1\lambda_1\mathbf x_1+c_2\lambda_2\mathbf x_2)\\=c_1^2\lambda_1||\mathbf x_1||^2+c_2^2\lambda_2||\mathbf x_2||^2\\=24c_1^2+8c_2^2=4$

$\implies6c_1^2+2c_2^2=1$