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Study Guides > College Algebra

Key Concepts & Glossary

Key Equations

General Form equation of a conic section Ax2+Bxy+Cy2+Dx+Ey+F=0A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0
Rotation of a conic section x=xcos θysin θy=xsin θ+ycos θ\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 θ,where cot(2θ)=ACB\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 Ax2+Bxy+Cy2+Dx+Ey+F=0A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0 where A,BA,B and CC are not all zero. The values of A,BA,B, and CC determine the type of conic.
  • Equations of conic sections with an xyxy term have been rotated about the origin.
  • The general form can be transformed into an equation in the x{x}^{\prime } and y{y}^{\prime } coordinate system without the xy{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(2θ)>0\cot \left(2\theta \right)>0, then θ\theta is between (0,45)\left(0^\circ ,45^\circ \right); if cot(2θ)<0\cot \left(2\theta \right)<0, then θ\theta is between (45,90)\left(45^\circ ,90^\circ \right); and if cot(2θ)=0\cot \left(2\theta \right)=0, then θ=45\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

Licenses & Attributions