Key Concepts & Glossary
Key Equations
General Form equation of a conic section | |
Rotation of a conic section | |
Angle of rotation |
Key Concepts
- Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
- A nondegenerate conic section has the general form where and are not all zero. The values of , and determine the type of conic.
- Equations of conic sections with an term have been rotated about the origin.
- The general form can be transformed into an equation in the and coordinate system without the term.
- An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section.
Glossary
- angle of rotation
- an acute angle formed by a set of axes rotated from the Cartesian plane where, if , then is between ; if , then is between ; and if , then
- degenerate conic sections
- any of the possible shapes formed when a plane intersects a double cone through the apex. Types of degenerate conic sections include a point, a line, and intersecting lines.
- nondegenerate conic section
- a shape formed by the intersection of a plane with a double right cone such that the plane does not pass through the apex; nondegenerate conics include circles, ellipses, hyperbolas, and parabolas