Finding Axes of Conic Section Equation

In summary, the conversation discusses the process of determining the X and Y axis of a general conic section equation. The suggested methods include finding the eigenvectors of the matrix (A B/2; B/2 C) and using the formula for rotating coordinates. The conversation also mentions the need to determine the eigenvalues of the matrix and how it relates to finding the values of a and b in the equation x^2 / a^2 - y^2 / b^2 = 1.
  • #1
aamirnshah
4
0
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
 
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  • #2
Welcome to PF!

Hi aamirnshah! Welcome to PF! :smile:

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. :wink:
 
  • #3


tiny-tim said:
Hi aamirnshah! Welcome to PF! :smile:

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. :wink:

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
 
  • #4
aamirnshah said:
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? :wink:
 

1. What is a conic section?

A conic section is a curve that is formed by the intersection of a cone with a plane. The three types of conic sections are the ellipse, parabola, and hyperbola.

2. How do you find the axes of a conic section?

To find the axes of a conic section, you need to first identify the type of conic section you are dealing with. The axes of an ellipse are the longer and shorter diameters of the ellipse, while the axis of a parabola is the line of symmetry. The axes of a hyperbola are the transverse and conjugate axes.

3. Can you find the axes of a conic section using its equation?

Yes, the equation of a conic section can provide information about its axes. For example, the coefficients of the x^2 and y^2 terms in the general equation of a conic section can indicate the type of conic section (ellipse, parabola, or hyperbola) and the orientation of its axes.

4. What is the significance of finding the axes of a conic section?

The axes of a conic section provide important information about its shape and orientation. They can help determine the eccentricity, foci, and vertices of the conic section, which are useful in applications such as astronomy, engineering, and physics.

5. Are there any shortcuts or tricks for finding the axes of a conic section?

Yes, there are a few tricks that can make finding the axes of a conic section easier. For example, for an ellipse or hyperbola, you can use the distance formula to find the distance between the foci and use that to determine the length of the axes. Additionally, knowing the properties of each type of conic section can help you quickly identify the axes and other important features.

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