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.<\/li>\n\t
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.<\/li>\n\t
Equations of conic sections with an [latex]xy[\/latex] term have been rotated about the origin.<\/li>\n\t
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.<\/li>\n\t
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.<\/li>\n<\/ul>
Glossary<\/h2>\n
angle of rotation<\/dt>
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]<\/dd><\/dl>
degenerate conic sections<\/dt>
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.<\/dd><\/dl>
nondegenerate conic section<\/dt>
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<\/dd><\/dl>","rendered":"
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.<\/li>\n
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.<\/li>\n
Equations of conic sections with an [latex]xy[\/latex] term have been rotated about the origin.<\/li>\n
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.<\/li>\n
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.<\/li>\n<\/ul>\n
Glossary<\/h2>\n
\n
angle of rotation<\/dt>\n
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]<\/dd>\n<\/dl>\n
\n
degenerate conic sections<\/dt>\n
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.<\/dd>\n<\/dl>\n
\n
nondegenerate conic section<\/dt>\n
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<\/dd>\n<\/dl>\n\n\t\t\t \n\t\t\t