Finding the center of conic sections

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May 15th 2008, 07:12 PM
Dergyll

Finding the center of conic sections

Hey guys, just ran across this question on a homework:

"Find the center of the conic section whose equation is: A x ²+B xy +C y ²+D x +E y +F=0"

How do I find them without any numbers in each term? Is there a general formula to solve for the center like -b/2a?

Any help is appreciated
Derg
May 15th 2008, 07:14 PM
Mathstud28

Quote:

Originally Posted by Dergyll

Hey guys, just ran across this question on a homework:

"Find the center of the conic section whose equation is: A x ²+B xy +C y ²+D x +E y +F=0"

How do I find them without any numbers in each term? Is there a general formula to solve for the center like -b/2a?

Any help is appreciated
Derg

This is messy but you can complete the square (Puke)
May 15th 2008, 07:22 PM
topsquark

Quote:

Originally Posted by Dergyll

Hey guys, just ran across this question on a homework:

"Find the center of the conic section whose equation is: A x ²+B xy +C y ²+D x +E y +F=0"

How do I find them without any numbers in each term? Is there a general formula to solve for the center like -b/2a?

Any help is appreciated
Derg

Define what you mean by "center." For example I have never heard of the term used for a parabola. (Though there is a point called the focus that might serve a similar function.)

If B = 0 then it's fairly easy. As Mathstud28 suggested you can complete the square on the x and/or y terms (as applicable) and simplify from there. If B is not 0 then you need to rotate the coordinate system such that the new form has the x'y' coefficient equal to 0.

The question is odd. There is no way to approach this until at least some of the coefficients are known.

-Dan
May 16th 2008, 02:23 AM
Dergyll

That is exactly word for word what the question says. The center (because of the fact that the graph is NOT a parabola since both x and y are both squared) should be that of an eclipse. But I still don't know how to solve for it!!! Does the problem want me to plug numbers in?

Thanks
Derg
May 16th 2008, 02:37 AM
mr fantastic

Quote:

Originally Posted by Dergyll

[snip]
The center (because of the fact that the graph is NOT a parabola since both x and y are both squared) should be that of an eclipse. But I still don't know how to solve for it!!!
[snip]

It makes sense to talk about the centre of a hyperbola too, you know - it's the intersection point of its asymptotes .......

Quote:

Originally Posted by Dergyll

[snip]
Does the problem want me to plug numbers in?

Thanks
Derg

Not being able to consult with the author of the question, who knows? Although, if that's what the question wanted you to do, shouldn't it say so ....?
May 16th 2008, 04:35 AM
topsquark

Quote:

Originally Posted by Dergyll

That is exactly word for word what the question says. The center (because of the fact that the graph is NOT a parabola since both x and y are both squared) should be that of an eclipse. But I still don't know how to solve for it!!! Does the problem want me to plug numbers in?

Thanks
Derg

What graph? And the equation you posted was in terms of unknown constants A, B, ... You didn't state any conditions on them so some of them may be zero and may be positive or negative. The equation you gave was the general form for a conic section: it could be anything.

-Dan
May 16th 2008, 05:27 AM
Soroban

Hello, Derg!

If you haven't been taught about Rotations,
. . this problem is inappropriate.

Quote:

Find the center of the conic section whose equation is:
. . . $Ax^2 + Bxy + Cy^2 + DX + Ey + F \;=\;0$

There is a "discriminant": . $\Delta \;=\;B^2-4AC\qquad \begin{Bmatrix}\Delta = 0 & \text{Parabola} \\ \Delta > 0 & \text{Hyperbola} \\ \Delta < 0 & \text{Ellipse} \end{Bmatrix}$

If there is no $xy$ -term $(B = 0)$ , it is a standard conic.
. . Its axes are parallel to the coordinate axes.

If $B \neq 0$ , the conic has been rotated through angle $\theta$

. . where: . $\tan2\theta \:=\:\frac{B}{A-C}$

We can "un-rotate" the graph with these conversions:

. . $\begin{array}{ccc} X &=& x\cos\theta - y\sin\theta \\ Y &=& x\sin\theta - y\cos\theta \end{array}$

We also have:

. . $A' \;=\;A\cos^2\!\theta + B\sin\theta\cos\theta + C\sin^2\!\theta$
. . $B' \;=\;0$
. . $C' \;=\;A\sin^2\!\theta - B\sin\theta\cos\theta + C\cos^2\!\theta$
. . $D' \;=\;D\cos\theta + E\sin\theta$
. . $E' \;=\;\text{-}D\sin\theta + E\cos\theta$
. . $F' \;=\;F$

We get a new equation: . $A'X^2 + C'Y^2 + D'X + E'Y + F' \;=\;0$

. . and now you can complete the square, etc.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Note: They always use the term "center".
. . . . .In a parabola, it refers to the vertex.