Rotation and systems of quadratic equations

Okay... Let me be frank. This section is going to make you cry, because it definitely made me. BUT! With my guidance you are going to understand this so easy! 

Let's start with the equation of an xy-plane.

Ax^2+Bx^2+Cy^2+Dx+Ey+F=0

The objective of rotation is to eliminate the xy term because it completely messes up our graphing process! We do so in e following steps.

1. Find the angle in which we rotate the graph through the equation..

    Cot 2 theta = A - C

                            B

2. Use the angle to find x and y, in the equations

    X= x' cos theta - y' sin theta

    Y= x'sin theta + y' cos theta

3. Substitute x and y into the original equation. ( we will also try to rearrange this into the equation of a parabola, ellipse, or hyperbola). 

Remember that the general second degree equation is 

A'(x')^2 + C'(y')^2 + D'x' + E'y' + F' = 0

4. We graph this new second degree equation. The angle of rotation is the angle we found earlier.

Here's an example of how to do a rotation