Approach to the foundations of the conic sections // The Parable

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In previous publications, sharing in a general way the conic sections, where considering as a section of a circular cone, in which the points of the conic are characterized according to their distances to two lines and deduces a large number of geometric properties from their characterization, under the geometric terms scheme, where the changes of angles depending on the location of the intersection are evident, within this type of conic sections we find: circles, ellipses, hyperbolas and parabolas, but in this opportunity fellow readers is for the Parable section.


A set consisting of points on a plane that are equidistant from a given fixed point and a fixed line on the plane is a parabola, the fixed point is the focus of the parabola, the fixed line is the directrix. Information consulted in Calculus: various variables by George Brinton Thomas, 2006.


The interesting thing about the Parabola as part of the conic sections, is in the plane it can be a geometric place of all the points, which for Q this is equidistant in a fixed point, like its focus (F), this in function of a line fix that is its guideline, independently, since this line does not contain a focus, complying with the mathematical model, d (Q, F) = d (Q, L). Said points in the plane for the parabola case, is defined by P (x, y) where the cuan this distance contains the focus, which in turn will be equal to the straight line distance in L.


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A representation of the conic sections // The Parable.


A representation of the conic sections // The Parabola, where it is evident (F), which represents the focus of the parabola and L as the direction of the parabola, points from the origin at V (0,0) the distance d (P, F) and the distance d (P, L), where P> 0 is considered for this case.


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A representation of the conic sections // The Parable, where the equality between the distances of d (P, F) = d (P, L) is evidenced.


These equations reveal the symmetry of the parabola about the axis and, the axis is called the axis of the parabola (a shortened form of "axis of symmetry"), the point where the parabola crosses its axis is the vertex, the vertex of the parabola x ^ 2 = 4py is at the origin, the positive number P is the focal length of the parabola. Information consulted in Calculus: various variables by George Brinton Thomas, 2006.


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Thanks to the contributions mentioned above, it is also important to share the following fellow readers, taking a section of the parable can explain its axis, including its guideline, in order to further enrich knowledge about this context, within the field of mathematics and the calculation which is combined with the geometric aspect or the analytical geometry to mention it in a way, since so simple that the line that passes through (F) focus, this is pedicular to the straight line, which is the guideline, this relation is the axis that is formed in a completely symmetrical geometric shape, the other interpretation which is given to the axis that is in the points of the parabola or that is contained in it, forms another axis creating a geometric shape of a vertex, that is why the relationship of the distances through the formation of these geometric axes, serve as a basis to better explain the distance of the guideline and the focus of the parabola as its definition establishes. on.


The importance of the application of the conic sections in the case of the parabola, taking into account the conic general equation Ax ^ 2 + By ^ 2 + CX + Dy + F = 0, is that in addition to its conceptualization it has applications in the field of physics, astrology and telecommunications, in the case of parabolic antennas, that's why its name is used by means of satellite channels, where they interpret these signals that affect its surface, where they reflect a directive, which allows feeding the focus of the parabola of the received signal, as part of a receiver, thanks to these he studied also served as a basis, to explain part of the theory of the law of reflection, where the intensities emitted by a ray of light are applied, up to the moment it touches or reaches its emission on a surface, creating symmetrical axes to the directive, in which in this case they form an angle for initial α or incidence angle and for β as the reflection angle emitted by the light ray, when forming this axis simé trico, is where it would represent the foci of light ray intensities.


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Apollonius (3rd century BC) the epicycle emerged; He also analyzed, without any reference to astronomy, the properties of the conic sections, such as the parabola, the ellipse, which had their importance, later, in the Kepler - Newton theory in the 17th century. Information consulted in Introduction to The Concepts and Theories of Physical Sciences by Gerald James Holton, Stephen G. Brush, 1996.


[1] - Analytical Geometry. An Introduction to Geometry by Ana Irene Ramírez Galarza, 2004.


[2] - Practical mathematics by Claude Irwin Palmer, Samuel Fletcher, 2003.


[3] -consulted in Introduction to The Concepts and Theories of Physical Sciences by Gerald James Holton, Stephen G. Brush, 1996.


[4] -Calculation: various variables by George Brinton Thomas, 2006.


[5] -Descriptive Geometry: A Descriptive Geometry Compendium for Technicians By B. Leighton Wellman, 1976.


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