I have an ellipse expressed as $(a+1)x^2+2abxy+(1+b)y^2=r^2.$ I found its center to be $(0,0).$ How can I express this in standard form $\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$ in the $xy$ plane?
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Try rotating the axes and shifting the origin. – Ishraaq Parvez May 23 '21 at 04:39
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1$(b+a+2+\sqrt{(4a^2+1)b^2-2ab+a^2})x^2/2+(b+a+2-\sqrt{(4a^2+1)b^2-2ab+a^2})y^2/2-r^2=0,$ which is not an ellipse for all $a,b.$ – Jan-Magnus Økland May 23 '21 at 05:22
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@Jan-MagnusØkland Thanks for the help. Could you tell me when it may fail to be an ellipse. What should I test? – Richard May 23 '21 at 06:25
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1Try $a=b=2$ and $r=1$ in e.g. geogebra. – Jan-Magnus Økland May 23 '21 at 06:28
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@Jan-MagnusØkland Ok I see what you mean. Could you pls add some steps on how you arrived at your solution. Its the form I wanted. – Richard May 23 '21 at 06:33
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1If you have an hour, watch this linear algebra video. – Jan-Magnus Økland May 23 '21 at 06:39