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## Homework Statement

For the conic, 5x

^{2}+4xy+5y

^{2}=9, find the direction of the principal axes, sketch the curve.

I found the eigenvalues as

3,7 but have no idea whether the 'new' equation is

3(x')

^{2}+7(y')

^{2}

or

7(x')

^{2}+3(y')

^{2}

is there a way to determine which 'way' it goes?

I took a guess and just continued using the first formula:

I found the eigenvectors by substituting the eigenvalues and got:

λ=3, V1 = (-1,1)

λ=7, V2 = (1,1)

I then thought the principal axes would therefore be:

1/√2 (-1,1) and 1/√2 (1,1)

yet the answer seems to indicate the principal axes as, (1,-1) and (1,1), why is that? I thought you had to normalise the vectors to find the principal axes..