- #1

- 39

- 0

## Homework Statement

Identify and sketch the graph of the quadratic equation

4x² + 10xy + 4y² = 9

## The Attempt at a Solution

We put it in the matrix form:

[tex]\begin{pmatrix} 4 & 5 \\

5 & 4 \\

\end{pmatrix}[/tex]

Now we find the eigenvalues:

Det(A – xI) = [tex]\begin{pmatrix} (4-x) & 5 \\

5 & (4-x) \\

\end{pmatrix}[/tex]

= x² – 8x – 9

= (x – 9)(x + 1)

eigenvalues are [tex]\lambda[/tex]1 = 9 & [tex]\lambda[/tex]2 = -1

From there, it's pretty simple solving:

[tex]\lambda1x'^2 + \lambda2y'^2 = 9[/tex]

My problem here is: How do I know which eigenvalue is which? It obviously makes quite a bit of difference to the final result. Nothing in my textbook says.