# I Quadratic equation of two variables

#### LagrangeEuler

$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$
is
(a) elipse when $B^2-4AC<0$
(b) parabola when $B^2-4AC=0$
(c) hyperbola when $B^2-4AC>0$
I found this in Thomas Calculus. However for some values of parameters $A=17$, $C=8$, $B=\sqrt{4 \cdot 17 \cdot 8}$, $D=E=0$, $F=20$ I got just straight line. Where is mistake?

#### erbahar

There is no mistake. That quadratic equation is the general equation of a conic section. (When you intersect a cone with a plane.) So in special cases you can get a line, intersecting two lines or even a single point. A circle is also possible. Just google "conic section", you'll understand what I mean...

#### LagrangeEuler

There is no mistake. That quadratic equation is the general equation of a conic section. (When you intersect a cone with a plane.) So in special cases you can get a line, intersecting two lines or even a single point. A circle is also possible. Just google "conic section", you'll understand what I mean...
Well look my example. In my post for $A=17$, $C=8$, $B=\sqrt{4 \cdot 17 \cdot 8}$, $D=E=0$, $F=20$ I got just straight line. And for that is $B^2-4AC=0$, and it is not parabola.

#### BvU

Homework Helper
Look at the first wikipedia picture:

and imagine shifting the purple plane to the right until it contains the origin. The 'conic section' is a line in that case

#### erbahar

Let me expand Thomas' classification then:

(a) elipse (or circle; or point) when B2−4AC<0
(b) parabola (or line) when B2−4AC=0
(c) hyperbola (or two intersecting lines) when B2−4AC>0

the cases in the paranthesis are called degenerate conic sections. (not sure this nomencleture applies to circle though)

#### LagrangeEuler

Here is curve for $A=17$, $C=8$, $B=\sqrt{4 \cdot 17 \cdot 8}$, $F=20$. These are two parallel lines.

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#### BvU

Homework Helper
That's funny -- you sure ? With that choice of parameters the first three terms form a square and those are usually non-negative, not -20

#### LagrangeEuler

That's funny -- you sure ? With that choice of parameters the first three terms form a square and those are usually non-negative, not -20
Sorry. My mistake. $F=-20$. Other parameters, are the same. So equation is
$$17x^2+\sqrt{4 \cdot 17 \cdot 8}xy+8y^2=20$$

#### BvU

Homework Helper
Sort of like $x^2 + 2xy +y^2 = 1$ if I try to keep it simple.
The intersection of $z^2=(x+y)^2$ with $z^2=1$ in homogenous coordinates with a determinant 0, leading to one of the degenerate cases

I grant you I have a hard time forming a picture like in #4 ... ( @erbahar ? )

#### erbahar

Sort of like $x^2 + 2xy +y^2 = 1$ if I try to keep it simple.
The intersection of $z^2=(x+y)^2$ with $z^2=1$ in homogenous coordinates with a determinant 0, leading to one of the degenerate cases

I grant you I have a hard time forming a picture like in #4 ... ( @erbahar ? )
Didn't quite understand the question (I feel like I might have missed the context, sorry)
My initial reply #2 was to give a geometrical answer to the question.

Of course you can always write something like:
$(ax + by + c)^2 = -4$

which is a perfetly legitimate quadratic, however it does have no solution in the real plane. Parallel line example is also of that sort.

eg. $(ax + by + c)(ax + by + d) = 0$

#### BvU

Homework Helper
OP corrected the value of F so the form is $(ax+by)^2 =$ positive, with two parallel lines. I wondered what picture to form of a plane intersecting a cone in that specific case ...

#### erbahar

OP corrected the value of F so the form is $(ax+by)^2 =$ positive, with two parallel lines. I wondered what picture to form of a plane intersecting a cone in that specific case ...
Oh, I see now. I don't think it's possible to picture that as an intersection.

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