## Key Equations

 General Form equation of a conic section $A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$ Rotation of a conic section $\begin{array}{l}x={x}^{\prime }\cos \text{ }\theta -{y}^{\prime }\sin \text{ }\theta \hfill \\ y={x}^{\prime }\sin \text{ }\theta +{y}^{\prime }\cos \text{ }\theta \hfill \end{array}$ Angle of rotation $\theta ,\text{where }\cot \left(2\theta \right)=\frac{A-C}{B}$

## 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 $A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$ where $A,B$ and $C$ are not all zero. The values of $A,B$, and $C$ determine the type of conic.
• Equations of conic sections with an $xy$ term have been rotated about the origin.
• The general form can be transformed into an equation in the ${x}^{\prime }$ and ${y}^{\prime }$ coordinate system without the ${x}^{\prime }{y}^{\prime }$ 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 $\cot \left(2\theta \right)>0$, then $\theta$ is between $\left(0^\circ ,45^\circ \right)$; if $\cot \left(2\theta \right)<0$, then $\theta$ is between $\left(45^\circ ,90^\circ \right)$; and if $\cot \left(2\theta \right)=0$, then $\theta =45^\circ$
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