To Conic Sections and Applications of Conic Sections
Introduction To Conic Sections
A curve that is obtained by the intersection of the surface of a cone with a plane is a conic section. The different types of conic sections include the parabola, hyperbola, ellipse, a circle – a special case of an ellipse. They are defined in the Euclidean plane consisting of distinguishing properties. This article also answers the question – how to identify conic sections from general form. The conic parameters include the following. Focus is defined as the set of points which is at a distance from a particular point and from a particular line is a directrix. The various types of conic sections are categorised through the value of eccentricity.
1] 0 < e < 1 – ellipse
2] e = 1 – a parabola
3] e > 1 – a hyperbola
4] e = 0 – circle
The line joining the midpoint that is the curve’s centre and foci of an ellipse or hyperbola are called the principal axis. Linear eccentricity is the distance between the focus and the centre. A chord passing through the focus and that is parallel to the directrix is the latus rectum. The half of a latus rectum is the semi-latus rectum. The distance from the focus and the appropriate directrix is defined as the focal parameter. The chord between the 2 vertices is the major axis and the minor axis is the shortest diameter of that of an ellipse. Its half-length if the semi-minor axis. The below table explains the different values of conic sections based on their equation and properties.
|Conic Section||Equation||Semi Latus Rectum||Focal Parameter|
|Circle||x2 + y2 = a2||a||∞|
|Ellipse||(x2 / a2) + (y2 / b2) = 1||b2 / a||b2 / √a2 – b2|
|Parabola||y2 = 4ax||2a||2a|
|Hyperbola||(x2 / a2) – (y2 / b2) = 1||b2 / a||b2 / √a2 + b2|
- If b2 − 4ac < 0 – ellipse equation
- if a = c and b = 0 – circle equation
- if b2 − 4ac = 0 – a parabola equation
- if b2 − 4ac > 0 – a hyperbola equation
- if a + c = 0 – a rectangular hyperbola equation
Applications of Conic Sections
1] Its major application is in the field of astronomy. The orbits of two huge objects interact following Newton’s law of universal gravitation representing conic sections provided their common centre is at rest.
2] In the design of radio-telescopes, optical telescopes and searchlights, the reflective properties of the conic sections are used.
3] The construction of a searchlight includes a parabolic mirror which acts as a reflector, a bulb at the focus. A similar design is incorporated in a parabolic microphone.
4] The path traversed by the planets around the sun is in the form of ellipses with the sun at the centre at one fixed point.
5] To converge light beams at the parabolic focus, parabolic mirrors are used.
6] Parabola finds its use in the design of car headlights and spotlights as it helps in concentrating the light beam.
7] The path of the objects thrown near the surface of the earth follows a parabolic path.
8] A navigation system named LORAN uses the concept of a hyperbola.
Conic sections find its application in most of the fields. A few of them are listed above for reference.