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How can convert the general form of ellipse equation in the standard form? $$-x+2y+x^2+xy+y^2=0$$ Thank you in advance?

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You really need to provide some context for this question if you are hoping that users here will be able to help you: What progress have you made so far? What exactly is causing you problems? Is the question homework? –  Old John Dec 6 '13 at 21:11

1 Answer 1

The idea is to rotate your axes , so that the $xy$ term disappears. Define a transformation ( a rotation ) $x'=rcos\theta, y'=rsin\theta$ , sub-in in your equation, and set the mixed $x'y'$-terms equal to $0$. This will give you the necessary angle of rotation to make the $xy$ terms disappear.

See this post: How to put $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ in canonical form for another example.

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Thank you for your guidance! if I want plot it, how can do this even without standard form? –  user113962 Dec 6 '13 at 20:56
    
Only way I can think of is: take standard form, draw it and then rotate back to the original form by the same angle $\theta$. But let me see if I can think of something else. –  user99680 Dec 6 '13 at 21:11
    
I would be grateful if you help help me in this case. –  user113962 Dec 7 '13 at 7:27
    
Sure; have you found the rotated form yet ? –  user99680 Dec 7 '13 at 7:45
    
yes, but it is difficult way to show it if I want change parameters and then I obtain the standard form for each changing! –  user113962 Dec 7 '13 at 7:55

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