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The general equation of conics of the form: $$ Ax^2+Bxy+Cy^2+Dx+Ey+F=0 $$

and it is said that for a parabola $A=0$ or $C=0$ or $AC=0$ and $Bxy$ is associated with the rotations.

But for the given conic, $16x^2+8xy+y^2-74x-78y+212=0$

It seems to be a parabola even though $A=16\neq0$ and $B=1\neq0$.

What am I missing here in the general form of conics ?

What about the case with other conics ?

$A\neq C$ and $AC>0$ for ellipse

$A\neq C$ and $AC<0$ for hyperbola, etc.

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That $A=0$ or $C=0$ is not quite right as you've noticed. The discriminant $\Delta=B^2-4AC$ equals $0$ for a parabola, as with your conic. Note that if $B=0$ then necessarily $A=0$ or $C=0$ in order for the discriminant to be zero.

Edit: Likewise for your other rules. Those are true only if $B=0$. The more general rule is $\Delta<0$ is an ellipse. $\Delta>0$ is a hyperbola. See here

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  • $\begingroup$ Is there another name besides $D$ that we can use for the discriminant, since $D$ is also what we're calling the coefficient of $x$? $\endgroup$ – Tanner Swett Jul 5 '17 at 17:37
  • $\begingroup$ @TannerSwett Yes. Thank you. $\Delta$ is a much better choice here. $\endgroup$ – sharding4 Jul 5 '17 at 17:39
  • $\begingroup$ @sharding4 thnx. what about the other conics, how do i differentiate between ellipse, hyperbola etc. how can i get the complete general equation when $B\neq 0$ $\endgroup$ – ss1729 Jul 5 '17 at 17:42

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