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What Is A Parabola?

The parabola is a concept that has very different meanings, but its mathematical definition is as follows:

In mathematics, a parabola is the locus of points in the plane that are equidistant from a fixed point (called a focus) and a fixed-line (called a directrix).

Therefore, any point of a parabola is at the same distance from its focus and its directrix.

Furthermore, in geometry, the parabola is one of the conic sections together with the circumference, the ellipse, and the hyperbola. That is, a parabola can be obtained from a cone.

In particular, the parabola is the result of cutting a cone with a plane with an angle of an inclination concerning the axis of revolution equivalent to the angle of the generatrix of the cone. Consequently, the plane containing the parabola is parallel to the generatrix of the cone.

Elements Of A Parabola

The characteristics of a parabola depend on the following elements:

Focus (F): it is a fixed point inside the parabola. The distance from any point on the parabola to the focus is equal to the distance from that point to the directrix of the parabola.

Directrix (D): it is a fixed-line external to the parabola. A point on the parabola has the same distance to the directrix as it is to the focus of the parabola.

Parameter (p): is the distance from the focus to the guideline.

Radius vector (R): is the segment that joins a point of the parabola with the focus. Its value matches the distance from the point to the directrix.

Axis (E): it is the line perpendicular to the directrix that passes through the focus and is the axis of symmetry of the parabola, in the graph below it corresponds to the ordinate axis (Y-axis). It is also said focal axis.

Vertex (V): is the point of intersection between the parabola and its axis.

Focal length: is the distance between the focus and the vertex, or between the directrix and the vertex. Its value is always equal to p/2.

Straight Side

The straight side of a parabola is the chord within the parabola that passes through the focus and is parallel to the directrix.

Equations Of The Parabola

The equation of a parabola is a type of quadratic function because it must always have at least 1 term squared. Also, the equation of a parabola depends on whether it is oriented horizontally or vertically.

Thus, in analytic geometry there are several ways of expressing a parabola mathematically: the canonical or reduced equation, the ordinary equation, and the general equation of the parabola.

Reduced Or Canonical Equation Of The Parabola

What differentiates the reduced or canonical equation from the other parabolic equations is that the vertex of the parabola is the origin of coordinates, that is, the point (0, 0).

The form of the reduced equation of the parabola depends on whether it is horizontal or vertical. Look at the following graphic representation where the four possible variants are shown:

Where p is the characteristic parameter of the parabola.

As you can see in the previous image, when the variable x is squared the parabola is vertical, on the other hand, when the variable y is squared the parabola is horizontal. On the other hand, the direction of the branches of the parabola depends on the sign of the equation.

Ordinary Equation Of The Parabola

We have just seen what the equation of the parabola is like when its vertex or center corresponds to the origin of coordinates (the reduced or canonical equation), but what is the equation of the parabola if the vertex is outside the origin?

When the vertex of the parabola is any point, we use the ordinary equation of the parabola, whose expression is:

Where the center or vertex of the parabola is the point V (x₀, y₀).

The above equation corresponds to the parabola that is oriented vertically, or in other words, the focal axis of the parabola is parallel to the Y-axis.

Similarly, to define a parabola oriented horizontally (its focal axis is parallel to the X-axis), we must use the following variant of the ordinary equation of the parabola:

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