I Quadratic equation of two variables

Quadratic equation
[tex]Ax^2+Bxy+Cy^2+Dx+Ey+F=0 [/tex]
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?
 

member659127

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...
 
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

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Look at the first wikipedia picture:
243062

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

member659127

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)
 
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

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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
 
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
[tex]17x^2+\sqrt{4 \cdot 17 \cdot 8}xy+8y^2=20[/tex]
 

BvU

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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 ? )
 

member659127

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

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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 ...
 

member659127

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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