Nondegenerate conic sections are the curves obtained by intersecting a plane with a double-napped cone, which do not degenerate into simpler forms. These include ellipses, parabolas, and hyperbolas.
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Nondegenerate conic sections can be classified as ellipses, parabolas, or hyperbolas based on the angle of intersection between the plane and the cone.
The general second-degree equation for nondegenerate conic sections is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
Rotation of axes can simplify the general equation of a conic section by eliminating the $Bxy$ term.
The discriminant $B^2 - 4AC$ determines the type of conic: ellipse ($<0$), parabola ($=0$), or hyperbola ($>0$).
Conic sections have important geometric properties such as foci, directrices, and eccentricity.
Review Questions
What are the three types of nondegenerate conic sections?
How does rotation of axes affect the equation of a nondegenerate conic section?
What does the discriminant $B^2 - 4AC$ indicate about a conic section?
A type of nondegenerate conic section formed when a plane intersects both nappes (opposite cones) at an angle steeper than that made by the side of the cone with its base.