how would you determine the eccentricity of the conic section when represented by the equation

Question

ax^2+cy^2+dx+ey+f=0
if the equation represents an ellipse and then when it represents a hyperbola.please help.

Answer 1

Put

x_1 = x

x_2 = y

Then the quadratic form is of the form

Sum over i and j of
M_{i,j}x_i x_j = constant.

Then, since this equation is invariant under orhtogonal transformations, we can write it in diagonal form. If you define new coordinates x' and y' that are in the direction of the two egienvectors, the equation becomes:

lambda_1 x'^2 + lambda_2 y'2 = constant

where lambda_1 and lambda_2 are the two eigenvalues of M.

Answer 2

I now see that there is no xy term in your expression, so the problem is trivial. You can simply complete the square, no need to diagonalize a matrix.