Considering an F point and a straight r in flat, the set that contains all the points whose distance to F is equal to the distance to r is called parable. point F is the focus of the parabola and can never be one of the points on the line r. Otherwise, the distance between F and r will always be equal to zero.
Below is an example of parable with the demonstration of its point F and the line r.
In elementary school, the parables are used only to represent geometrically. high school functions. In high school, they are also the result of studies of the conical, in Analytical Geometry.
Elements of a parable
There are five main elements of the parable. They are geometric figures that receive special names due to their function and their importance in defining parables. Are they:
The) Focus
It is the F point used for the definition of the parable.
B) Guideline
And the straight r, also used in the definition of the parable. Remember that the distance between any point on the parabola and the line r is the same distance as that same point and its focus.
ç) Parameter
O parameter of a parable is the distance between your focus and yours guideline. This distance is the length of the line segment that connects the focus and the guideline, forming a right angle with it. To find this value, you can use the distance between point and line.
d) Vertex is the point of parable which is closest to your guideline. One of the properties of this point is that its distance until the focus of the parable is equal to half of the parameter. We can also say that the distance between this point and the guideline of the parabola is equal to half the parameter.
be the measure of the parameter of a parable represented by the letter p, the measurement of the VF segment will be given by:
FV = P
2
and) Axleinsymmetry
O axleinsymmetry of a parable is a straight line perpendicular to guideline that goes through your vertex. Consequently, this line also passes through the focus of the parabola and contains the segment called parameter.
The following image shows each of the elements of a parable:
Reduced equations of the parabola
there are two equations reduced from parable:
y2 = 2px
and
x2 = 2py
These equations are obtained by placing the vertex of a parable at the origin of a Cartesian plane. First, suppose the guideline of this parabola is parallel to the y axis of the plane, as shown in the following image.
Choosing any point P(x, y) na parable, we will have the following hypotheses:
1 – F coordinates: as the segment VF = p/2, then the coordinates of F are (p/2, 0). To see this, note that the x-axis in this construction is the axleinsymmetry gives parable.
2 – Coordinates of A: point A belongs to guideline, and the distance from P to A is equal to the distance from P to F. So, changing the position of point P, we will always have this characteristic. The coordinates of A are: (– p/2, y).
This is because A will always be at the same height as P, and its distance from the y axis is the same as the distance from V to F, with the sign inverted.
3 –The distance from P to A is equal to the distance from P to F, as this is the definition of parable.
Given these hypotheses, we can calculate the following equation, replacing it with the coordinates of each of the points P, A and F:
The second equation gives parable it has its calculations and constructions done in an analogous way to these, however, it presents the guideline parallel to the x axis.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-parabola.htm
