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Conic Sections

Orbits of comets Orbits of comets

The orbits of comets in solar system provides an interesting application of conic sections. Prior to 1970, 610 comets were identified, of which 245 had elliptical orbits, 295 had parabolic orbits and 70 had hyperbolic orbits. For example, Encke's comet has an elliptical orbit, and reappearance of this comet can be predicted every 3.3 years. The center of the sun is a focus of each of these orbits, and each orbit has a vertex at the point where the comet is closest to the sun, as shown in above figure.

If x is the distance between the vertex and the focus (in meters) and v is the speed of the comet at the vertex (in m/s), then the type of orbit is determined as: and , where M is the mass of the sun and G is the universal gravitational constant.

Introduction to Conics

Conic sections were discovered during the classical Greek period, 600 to 300 B.C. The conic sections are figures that can be formed by the intersection of a plane with a cone. There are four basic conic sections: circles, ellipses, parabolas and hyperbolas.

A conic section can be formally defined as: the set of points in a plane whose distances from a fixed point bear a constant ratio to the corresponding perpendicular distances from a fixed straight line. The fixed point is called the focus and the fixed straight line is called the directrix of the conic section. The constant ratio is called the eccentricity of the conic section and is denoted by e.

When the eccentricity is 0, that is, e = 0, the conic section is called a circle;
When e = 1, the conic section is called a parabola;
When e < 1, the conic section is called an ellipse;
When e > 1, the conic section is called a hyperbola.

The general equation for all conic sections is: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 with A, B, C not all zero.

Lighting the Olympic flame Lighting the Olympic flame For Olympic games, the Olympic flame acts as a symbol. It is traditionally lit at Olympia, Greece, using a parabolic mirror concentrating sunlight, and is then transported to the venue of the Games.

The set of all the points in a plane that are equidistant from the fixed line and a fixed point, not on the line, is called a parabola. The fixed point is called the focus and the fixed line is called the directrix of the parabola. A parabola is also formed by intersecting the plane through the cone and the top of the cone.

Parabolic antenna Parabolic antenna Parabolic antenna uses a parabolic reflector, which is a reflective surface used to collect or project energy such as light, sound, or radio waves. Dish antenna/parabolic dish is the most common form and is shaped like a dish as its name implies. One of the important advantages of a parabolic antenna is that, it has high directivity. The principle of working is similar to a search light or flashlight reflector to direct the radio waves in a narrow beam, or receive radio waves from one particular direction only.
Terms Related to Parabola
Vertex and Axis: The midpoint between the focus and the directrix is called vertex of the parabola. The line passing through the focus and the vertex is called axis of the parabola.
Chord and Focal chord: The line segment joining any two points on the parabola is called chord of the parabola. The chord which passes through the focus is called focal chord of the parabola.
Latus rectum: The focal chord perpendicular to the axis is called the latus rectum.
Double ordinate: A straight line drawn perpendicular to the axis of a parabola and terminated at both ends by the curve is called the double ordinate.
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