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I don't understand the importance of putting the standard equation -which I find straight to the point- to the General form and driving the properties from it whereas we could them simply from the standard form.

I'd appreciate cases with examples.

edit: based on my college textbook the equation of the circle per example is given standerd form: $\ r^2 = ( x - h ) ^ 2 + ( y - k ) ^ 2\ $ , center is ( h,k )

then another " general " form : $ \ x ^ 2 + y ^ 2 - 2gx -2 f y + c \ $ where c = $ \ h^2 + k^2 - r ^2 \ $ and the new center is ( - g, -f ) and obviosly some substitutions to fill in. i don't understand the point of such " simplification "

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  • $\begingroup$ Do you mean, "... simply from the general form"? $\endgroup$ – Henning Makholm Mar 20 '17 at 15:35
  • $\begingroup$ I do yes, edited. $\endgroup$ – sarah Mar 20 '17 at 15:43
  • $\begingroup$ what do you means for ''general'' and ''standard''? $\endgroup$ – Emilio Novati Mar 20 '17 at 15:45
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The usual way to derive the properties of the conic sections in analytic geometry is to work with their standard forms. The standard forms have been chosen such that they are much easier to work with.

The reason general forms are taught is not that you're supposed to start with an equation in standard form and then convert it to the general form -- that is usually not a very enlightening thing to do (though there are a few specific contexts where this can be a useful operation).

The point is that you can go in the opposite direction. If you're trying to model some particular situation and -- after a torrent of algebra, simplifying everywhere and collecting like terms, etc. etc. -- you end up with something of the form $$ Ax^2+Bxy+Cx^2+Dx+Ey+F = 0 $$ you should be able to recognize, "oooh, this is the general form of the equation for some conic secion". Then you can start working out which conic section it is and what the standard form of its equation is. Then, knowing the standard form will tell you what is the shape of the solution set to the original equation.

In order to figure out how to go from general to standard, you need to know how the general form of the various conic sections look like -- and that is the reason textbook will derive the general forms in the first place.

For example, the general equation above determines a parabola (or one of the degenerate cases) exactly if $B^2=4AC$ -- but how did we discover that fact? An easy way is to take the standard parabola $x^2=y$ and investigate what this equation turns into when we scale, turn, and translate it.

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In real life, the conics appear in their general form, with up to six coefficients.

As it is difficult to discuss the properties in this setting, some of the coefficients are eliminated by means of rigid, similarity or affinity transforms (translation, rotation, isotropic scaling and/or anisotropic scaling), leading to the simplified equations.

This only works for properties that are preserved by the corresponding transforms.

For instance, any ellipse can be transformed in the unit circle, and the transformed tangents to the ellipse are tangent to the circle; on the opposite, the area is not preserved...

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