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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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    $\begingroup$ First of all, decide whether it is an ellipse, a hyperbola, or a parabola. This is easy. $\endgroup$ – Siminore Sep 14 '12 at 12:06
  • $\begingroup$ May have a look into : math.stackexchange.com/questions/194535/ellipse-question $\endgroup$ – lab bhattacharjee Sep 14 '12 at 12:23
  • $\begingroup$ @Siminore, yes it is easy! :) But then.. what do I have to do? I'd like to understand the steps needed to get the canonical form. Could you help me? $\endgroup$ – sunrise Sep 14 '12 at 17:41
  • $\begingroup$ @labbhattacharjee I'm sorry, thank you, but my Prof doesn't like this proceeding... $\endgroup$ – sunrise Sep 14 '12 at 18:01
  • $\begingroup$ @sunrise What are you allowed to use, then? $\endgroup$ – Siminore Sep 15 '12 at 8:05
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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.

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