Key Equations

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

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