A conic is a curve obtained as the intersection of the surface of a cone with a plane. The primary types of conics are ellipses, hyperbolas, and parabolas.
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Conics can be classified into three main types: ellipses, hyperbolas, and parabolas based on the angle at which the plane intersects the cone.
The general equation for a conic section in Cartesian coordinates is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
An ellipse has a positive discriminant $(B^2 - 4AC < 0)$, while a hyperbola has a negative discriminant $(B^2 - 4AC > 0)$. A parabola has zero discriminant $(B^2 - 4AC = 0)$.
In polar coordinates, conics can be described by equations of the form $r = \frac{e \cdot d}{1 + e \cos \theta}$ or $r = \frac{e \cdot d}{1 + e \sin \theta}$ where $e$ is the eccentricity and $d$ is the directrix distance.
Eccentricity ($e$) determines the shape of a conic: for ellipses $0 < e < 1$, for parabolas $e = 1$, and for hyperbolas $e > 1$.
Review Questions
What is the general equation of a conic section in Cartesian coordinates?
How does eccentricity affect the shape of different types of conic sections?
Write the polar coordinate equation for an ellipse.
Related terms
Ellipse: A type of conic section formed when a plane intersects a cone at an angle less than that made by its side; characterized by two foci.
Hyperbola: A type of conic section formed when a plane intersects both nappes (sides) of a double cone; characterized by two branches and two foci.
Parabola: A type of conic section formed when a plane is parallel to one side of the cone; characterized by having only one focus.