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I do not get why we would like to have a general form of the ellipse when the standard form tells us all the information we need about it, whereas the general does not -> we'll have to convert it in order to extract the meaning.

One possible reason that comes into my mind is that nature gives us the equation in general form and the standard form is just kind of a luxury that we have built in order to make things more transparent and obvious.

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    $\begingroup$ Probably for the same reason we like to have a general equation $ax+by+c=0$ of the line, even if the standard form $y=0$ tells us all the information we need about it. $\endgroup$ May 29, 2022 at 6:36
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    $\begingroup$ The standard form is nice if all you care about is the ellipse. But what if you have two ellipses lying in some weird angle wrt each other? Then at most one of them can be in standard form. $\endgroup$ May 29, 2022 at 6:55
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    $\begingroup$ The question you are asking can be generalized: why do we use general matrices whereas diagonal matrices are so easy :) ? Asking it under the form easy/difficult is not the real issue, the important thing that you will find in all mathematics is classification into "canonical" forms through a change of basis (or change of reference in general) helping to compare for example 2 ellipses wrt a certain criteria : do they have the same ratio b/a , etc. ? or two vector spaces : do they have the same underlying space $\mathbb{R^n}$ (i.e., the same dimension), etc... $\endgroup$
    – Jean Marie
    May 29, 2022 at 7:13
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    $\begingroup$ What exactly do you mean by general form and standard form? Being specific there should make your question easier to answer consistently. $\endgroup$
    – MvG
    May 29, 2022 at 19:38
  • $\begingroup$ You can get lots of meaning from the equation of an ellipse in general form -- whether a given point is inside or outside, whether a given line intersects it (and what the points of intersection are), the points of tangency with lines of a given slope (and the equations of those lines) -- all without knowing the orientation or length of the axes. Sometimes the standard form is just not necessary. $\endgroup$
    – David K
    May 30, 2022 at 1:20

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