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CONIC SECTIONS

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A conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conconic_sec_1ical surface) with a plane. A conic may be defined as a plane algebraic curve of degree 2. It can be defined as the locus of points whose distances are in a fixed ratio to some point, called a focus, and some line, called a directrix.


Relatedconic_sec_2 Terms

The point V is called the vertex; the line l is the axis of the cone. The rotating line m is called a generator of the cone. The vertex separates the cone into two parts called nappes.


If we take the intersection of a plane with a cone, the section so obtained is called a conic section. Thus, conic sections are the curves obtained by intersecting a right circular cone by a plane.



Sections of a cone

We have the following sections of cone:

•    Circle

•    Ellipse

•    Parabola

•    Hyperbola


Each of the geometric figures are obtained by intersecting a double-napped right circular cone with a plane. Thus, the figures are called conic sections or conics.


Let’s discuss briefly about each of the above sections of cone:


conic_sec_3Circle

When β = 90°, the section is a circle.


If the plane cuts completely across one nappe of the cone and is perpendicular to the axis of the cone, the curve of the section is called a circle.



Ellipseconic_sec_4

When α < β < 90o, the section is an ellipse.

If the plane isn't perpendicular to the axis of the cone, it is called an ellipse.


An ellipse is the set of all points in a plane, the sum of the distances from two fixed points in the plane is constant. Many comets have elliptical orbits.



conic_sec_5Parabola

When β = α; the section is a parabola

If the plane doesn't cut across one entire nappe or intersect both nappes, the curve of the intersection is called a parabola.


A parabola is the set of all points in a plane equidistant from a fixed point and a fixed line in the plane.



Hyperbolaconic_sec_6

When 0 ≤ β < α; the plane cuts through both the nappes and the curves of intersection is a hyperbola.

If the plane cuts through both nappes of the cone, the curve is called a hyperbola.


The hyperbola is the set of all points in a plane. The difference of whose distance from two fixed points in the plane is the positive constant.



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