Key Equations
| General Form equation of a conic section | [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[/latex] |
| Rotation of a conic section | [latex]\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}[/latex] |
| Angle of rotation | [latex]\theta ,\text{where }\cot \left(2\theta \right)=\frac{A-C}{B}[/latex] |
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