# Conic sections question

1. Nov 4, 2006

Here's the question:
Consider the equation:
Ax^2+Cy^2+Dx+Ey+F=0

Consider the cases AC>0, AC=0 and AC<0 and show that they lead to an ellipse, parabola and hyperbola respectively, except for certain degenerate cases. Discuss these degenerate cases and the curves that arise from them

Don't really know where to start. I can 'prove' it by doing examples but that is not sufficient. Can somebody get me started on answering this question please?
Thanks

2. Nov 4, 2006

### arildno

Complete the squares, and rearrange in the cases where neither A or C are zero.
Then tackle the cases where either A, C or both are zero.

3. Nov 4, 2006

Ok... so I complete the square to:
EDIT: I suck at using LaTeX. Give me a minute...

Ok here it is:
http://img296.imageshack.us/img296/4413/eqnyo7.jpg [Broken]

Last edited by a moderator: May 2, 2017
4. Nov 4, 2006

### arildno

In the case of non-zero A and C, it is simpler to do it the following way, in order to not get into the silly trouble of square-rooting negative numbers:
$$A(x+G)^{2}+C(y+H)^{2}=I, G=\frac{D}{2A}, H=\frac{E}{2C}, I=AG^{2}+CH^{2}-F$$

Last edited: Nov 4, 2006