Parabola . Parabola is the locus of points on a plane that are equidistant from a line (axis or directrix) and a fixed point called focus.
Summary
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- 1 Definition
- 2 Properties
- 3 Equation
- 1 Breakpoints
- 4 Applications
- 5 Sources
Definition
The parabola is the curve obtained as a result of cutting a straight cone with a plane parallel to its generatrix. Its fundamental elements are shown in the image.
Point F is called the focus and line d is the directrix of the parabola.
The line that passes through the focus and is perpendicular to the directrix is called the axis of the parabola.
In the figure above, the axis of the parabola coincides with the Y axis. The point at which the axis intersects the parabola is called vertex V and is the point whose distance from the directrix is minimum.
The distance between the vertex and the focus is known as Focal Distance or Focal Radius.
Properties
- It is an open curve, it consists of two curve arcs (branches) without common points that extend indefinitely.
- It has two perpendicular axes of symmetry; therefore it is centrally symmetrical and has a center.
- A line of symmetry does not contain points on the curve
Equation
The equation of a parabola is y – y 0 = a (x – x 0 ) 2 . This equation is the parabola with vertical axis and whose vertex is (x 0 , y 0 ) .
Cut points
The cut points of the parabola with the axis of the abscissa , if any, are those that have the second coordinate y = 0 and the first coordinates are the solutions of the quadratic equation a (x – x 0 ) 2 + y 0 = 0 (obtained by requiring the condition y = 0 ).
The cut-off point with the ordinate axis , there is only one, is the one with the first coordinate x = 0 and the second coordinate y = a · x 0 2 + y 0 (obtained by requiring y = 0 ).
Applications
The parabola finds its application in many branches of applied science, because the graphs of quadratic equations are parabolas. For example, the ideal path of body motion under the influence of gravity.
It is also used in the construction of satellite antennas, taking advantage of the principle that a parabolic mirror reflects the rays emitted from its focus in parallel, as well as radio telescopes, which are also based on the concentration of the received signals.
The concentration of solar radiation at a point using a parabolic reflector is used in solar cookers.
