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[[File:Conic_Sections.svg|Conic Sections|thumb]] [[File:Conic_section_with_torch_light.svg|Conic section with torch light|thumb|left]] [[File:Conics-Converge-Diverge.jpg|Conics-Converge-Diverge|thumb|left]] [[File:TypesOfConicSections.jpg|TypesOfConicSections|thumb]] [[Image:Eccentricity.png|Eccentricity|thumb]] '''Conic sections''' are the curves obtained as the intersection of the surface of a [[cone]] with a [[plane]]. The four basic types of conic sections are the [[parabola]], [[ellipse]], [[circle]], and [[hyperbola]]. These shapes have been studied since ancient times and have important applications in [[mathematics]], [[physics]], [[engineering]], and many other fields.

==Definition==
A conic section can be defined as the locus of all points \(P\) such that the distance from \(P\) to a fixed point, called the [[focus]], is a constant multiple of the distance from \(P\) to a fixed line, called the [[directrix]]. The constant ratio is called the eccentricity (\(e\)), and it determines the type of conic section:
* If \(e=0\), the conic is a [[circle]].
* If \(e<1\), the conic is an [[ellipse]].
* If \(e=1\), the conic is a [[parabola]].
* If \(e>1\), the conic is a [[hyperbola]].

==Equations==
The general quadratic equation in two variables \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), where \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) are constants, represents a conic section. The nature of the conic section can be determined by the discriminant \(B^2 - 4AC\):
* If \(B^2 - 4AC < 0\), the equation represents an ellipse or a circle.
* If \(B^2 - 4AC = 0\), the equation represents a parabola.
* If \(B^2 - 4AC > 0\), the equation represents a hyperbola.

==Applications==
Conic sections have numerous applications across various fields:
* In [[astronomy]], the orbits of planets and comets are often described by conic sections, with the [[Sun]] at one of the foci.
* In [[optics]], mirrors shaped like parts of a parabola can focus parallel rays of light to a single point, and ellipsoidal mirrors can focus light from one point to another.
* In [[architecture]] and [[engineering]], the principles of conic sections are used in the design of structures such as bridges, domes, and arches for their aesthetic appeal and structural efficiency.

==History==
The study of conic sections can be traced back to ancient [[Greece]], where mathematicians like [[Euclid]] and [[Apollonius of Perga]] laid the foundational work. Apollonius's work, "Conics," significantly advanced the understanding of these curves, introducing terms such as ellipse, parabola, and hyperbola.

==See Also==
* [[Algebraic geometry]]
* [[Analytic geometry]]
* [[Orbital mechanics]]

[[Category:Mathematics]]
[[Category:Geometry]]
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Conic section - Revision history

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