how to know the axes of an ellipse after rotation.

Question

I came around a question:

$P=\begin{bmatrix}3 & 1\\1 & 3\\ \end{bmatrix}$. Consider the set S of all vectors $\begin{pmatrix}x\\y\end{pmatrix}$ such that $a^2+b^2=1$ where $\begin{pmatrix}a\\b\end{pmatrix}=P\begin{pmatrix}x\\y\end{pmatrix}$ Then S is ?

The answer to the above is : an ellipse with major axis along $\begin{pmatrix}1\\1\end{pmatrix}$

I solved the above and got an equation of a rotated ellipse : $10x^2+10y^2+12xy=1$ and with Matrix_representation_of_conic_sections I got the below matrix for the resulting ellipse

$\begin{bmatrix} 10 & 6\\ 6 & 10\\ \end{bmatrix}$

Solving it I get the eigen vales as 16,4 and one of the vector (for eigen value 16) $\begin{pmatrix}1\\1\end{pmatrix}$. Now my question is: Is this the vector the answer is talking about? If yes, how to determine if its along major or minor axis ?

Answer 1

Apparently this very section of wiki clears my doubt:

By the principal axis theorem, the two eigenvectors of the matrix of the quadratic form of a central conic section (ellipse or hyperbola) are perpendicular (orthogonal to each other) and each is parallel to (in the same direction as) either the major or minor axis of the conic. The eigenvector having the smallest eigenvalue (in absolute value) corresponds to the major axis.

The answer which mentioned above was wrong (probably a misprint in the book I was following).

The correct answer should be : an ellipse with minor axis along $\begin{pmatrix}1\\1\end{pmatrix}$ as the eigen value 16 has the largest value so it should correspond to minor axis.