Finding Axes of Conic Section Equation

[aamirnshah]

Hi,

I've been stuck on this for a few days now.

I am trying to determine the X and Y axis of the general conic section equation:

Q(x,y)=Ax^2 +Bxy+Cy^2 +Dx+ Ey+F =0

I have A, B, C, D, E, and F.

I have already determined the center coordinates, but I have not found information on how to find the two minor and major axes of the equations.

Any help is appreciated. Thanks

AS

 

[tiny-tim]

Welcome to PF!

Hi aamirnshah! Welcome to PF!

They'll be the eigenvectors of the matrix (A B/2; B/2 C).

Alternatively, just apply the formula for rotating coordinates through a particular angle to the original equation so as to produce an xy term of zero.

 

[aamirnshah]

Hi aamirnshah! Welcome to PF!

They'll be the eigenvectors of the matrix (A B/2; B/2 C).

Alternatively, just apply the formula for rotating coordinates through a particular angle to the original equation so as to produce an xy term of zero.

Wow thanks for the quick reply. So I have obtained the eigenvector of the matrix (A B/2; B/2 C). From my research I have determined that I also need to determine the eigenvalues of that matrix. But from here, I am still confused. How does this relate to knowing the 'a' and the 'b' in the equation:

x^2 / a^2 - y^2 / b^2 = 1

 

[tiny-tim]

From my research I have determined that I also need to determine the eigenvalues of that matrix. But from here, I am still confused. How does this relate to knowing the 'a' and the 'b' in the equation:

x^2 / a^2 - y^2 / b^2 = 1

Hint: what are the eigenvalues associated with x²/a² - y²/b² = 1?