General Rotated Ellipse/Hyperbola

The general form of a rotated ellipse or hyperbola can be given by:

(x - h)² + B(x - h)(y - k) + C(y - k)² = D

As usual (h, k) is the center of the object.

"B" controls the rotation, while C reflects the ratio of the lengths of sides of the bounding box. D is actually determined by both the rotation and the bounding box

Play with the Upper Left and Lower Right corners of the bounding box, and use the slider to rotate the object by changing the value for "B." See if you can find a pattern between the values of "B" and "C" that will tell you when the object is an ellipse, and when it is an hyperbola.

If you're familiar with calculus, think about how you'd find the four points of tangency of the bounding box and the conic section.

Dave Matthews, Created with GeoGebra