I Differential equation of all the conics in the plane

[Michii]

Hi, the problem is parametric families:
To find Differential equation of all the conics in the plane with the origin in the center
But when you speak of center at the origin being the equation of the conics: Ax ^ 2 + Bxy + cy ^ 2 + Dx + ey + F, is it correct to take the origin by making x and y equal to 0? Or what exactly does it refer to?

 

[haruspex]

Hi, the problem is parametric families:
To find Differential equation of all the conics in the plane with the origin in the center
But when you speak of center at the origin being the equation of the conics: Ax ^ 2 + Bxy + cy ^ 2 + Dx + ey + F, is it correct to take the origin by making x and y equal to 0? Or what exactly does it refer to?

I'm not sure what the question means. Where is the centre of a parabola? For anything else, you could look for a point about which the curve is symmetric.

 

[Michii]

Sorry I was not very clear, the question is: what does it mean that the equation of the conic has the origin in the center ?, I think that it would only be to equalize to zero the equation of the conic thus: Ax ^ 2 + Bxy + cy ^ 2 + Dx + ey + F = 0, thanks

 

[haruspex]

Sorry I was not very clear, the question is: what does it mean that the equation of the conic has the origin in the center ?, I think that it would only be to equalize to zero the equation of the conic thus: Ax ^ 2 + Bxy + cy ^ 2 + Dx + ey + F = 0, thanks

No, that is a completely general equation for a conic. It is not normalised in any way.
As I wrote, for most conics you could argue that the centre is the point of maximum symmetry. Making that the origin should give you an equation with fewer parameters. (How many fewer, do you think?)
But I cannot make this work for a parabola. It has no natural centre.

Edit: also does not work for the degenerate case of a single straight line.

 

[Michii]

Thank you very much for your help!

 

[haruspex]

Thank you very much for your help!

Did you figure out the symmetry?