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## #1 2009-12-07 07:57:11

ccmint
Member
Registered: 2009-10-11
Posts: 22

### How is the conic section discriminant derived?

Hi,

How can I derived the conic section discriminant from the general equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

The book I'm reading says that it's taken from A,B,C. I don't see how Ax^2 + Bxy + Cy^2 can be turned into b^2-4ac. I know that Ax^2 + Bx + C can be used to get the discriminant from learning about quadratics, but like I said the book says it uses A,B,C. My best guess is that they meant it's derived from Ax^2 + Dx + F. I'm not sure but I think I'm interpreting the text wrong. Can someone please clear this up for me? Thanks.

Last edited by ccmint (2009-12-07 09:16:03)

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## #2 2009-12-07 18:53:29

mikau
Member
Registered: 2005-08-22
Posts: 1,504

### Re: How is the conic section discriminant derived?

I don't see how Ax^2 + Bxy + Cy^2 can be turned into b^2-4ac.

Its easy! Watch, if we have 5x^2 + 3xy + 2y^2 and we want to find the discriminant, we observe A=5, B=3, and C=2, so to find B^2 -4*AC we just  substitute:
(3)^2 - 4*(5)*(2)= 9-40=-31.

Last edited by mikau (2009-12-07 18:54:10)

A logarithm is just a misspelled algorithm.

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## #3 2009-12-08 07:47:35

ccmint
Member
Registered: 2009-10-11
Posts: 22

### Re: How is the conic section discriminant derived?

Hi Mikau, thanks for your answer, but what I'm trying to say is how was the formula B^2 - 4AC derived from Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0? Where did B^2 - 4AC come from?

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## #4 2009-12-10 19:04:52

mikau
Member
Registered: 2005-08-22
Posts: 1,504

### Re: How is the conic section discriminant derived?

I don't know much about the significance of the descriminant in the context of the general conic equation, but its likely an advanced topic that you'd learn in a later course.  Right now, they're just telling you that B^2-4AC is defined to be the descriminant, you don't need to know where it came from right now. Often in math you learn how to use something before learning where it came from or why it works. And some things you can use without ever knowing exactly how they work. However! It is always good to ask questions and try to understand everything you can. :]

A logarithm is just a misspelled algorithm.

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## #5 2009-12-11 08:38:22

ccmint
Member
Registered: 2009-10-11
Posts: 22

### Re: How is the conic section discriminant derived?

Yup, I was just curious. Thanks for your feedback. I think I figured it out, I'm teaching myself this stuff. I had a pretty crummy math education. Teachers weren't that great. Anyways, from your reply on my other question "How to derive other equations from the general conic section equation?" I'm assuming that Bxy = 0, Cy2 = 0, and Ey = 0 for the general conic section equation: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 and what's left is a quadratic. From that I can get the discriminant.

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