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I have $x^2+2xy-2y^2+x-4y=0$ and I have to find its canonical form, but I'm a little confused.. I'd like to understand very well what I have to do.. Can you help me, please? Thanks!

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    @Siminore I can only use translations + rotations.. I can't use the derivation of rotation formula. I have to find the centre of the conic and then, using the equations set $\begin {cases}x=X+ \alpha \\ y=Y+\beta \end{cases}$ $(\alpha, \beta$ are center coords), I have to substitute these x and y in the equation of the conic. Then, I know that I have to diagonalize a matrix but.. which matrix? and then? What do I have to do?2012-09-15

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You want to eliminate the term involving $xy$. The simplest way for this example is to notice that $x^2+2xy=(x+y)^2-y^2$. So we use new variables $X=x+y, Y=y$ or $x=X-Y, y=Y$. Making this substitution gives $X^2-3Y^2+X-5Y=0$, and so it is a hyperbola.