Introduction

To understand the two-dimension standard forms of a parabola, it is essential to know what parabolas are. A parabola is produced due to the intersection of a right circular cone and a plane parallel to a part of the cone. It is usually described as a conic section. It is also a graph of an equation that includes a set of points in a plane arranged in a certain way. Let us look into a parabola’s meaning, properties and equations. These equations of a parabola are essential to solve various mathematical problems.

What do you mean by parabola? 

Before understanding two-dimension standard forms of parabola, let us first understand what parabolas are. A parabola is defined as an equation of a set of points in a plane in which all the points of a curve are equidistant from the given fixed point and a fixed-line. The fixed point referred to here is known as the parabola’s focus and is represented by the symbol fixed point ‘F’, and the fixed-line is known as the directrix of the parabola. The vertex of the parabola is halfway between the directrix and the focus. The parabola is usually U-shaped, and it is symmetric on either side of the axis. 

Equation of a parabola 

The general equation of a parabola is represented depending on the two-dimension standard forms of the parabola. In the cases where the parabolas have the vertical axis, then the general equation of a parabola is written as (x – h)2 = 4p(y – k), in which p≠ 0. Here, the given vertex of the parabola is (h, k). Now the focus for this equation is located at the point (h, k + p), then the directrix of this equation is y = k – p. If the p is greater than 0, then the parabola faces or opens upwards, and if p is less than 0, then the parabola faces or opens downwards. In case the parabola has a horizontal axis, the equation is (y – k)2 = 4p(x – h), where p≠ 0. Here the vertex is (h, k), the focus is at (h + p, k), and the directrix is represented by the line x = h – p. These are the general or standard forms of a parabola. 

Types of standard equations

Standard equations of the parabola are essential in solving certain mathematical equations and problems. Two-dimension standard forms of parabola display various equations based on their vertex focus etc.

The first equation is y2 = 4ax. In this equation, x (positive axis) is the parabola’s axis, and the focus is represented as (a,0). The directrix of this parabola is x = -a.

1. The second equation is y2 = -4ax. In this equation, the axis of the parabola is x (negative axis) is the parabola’s axis. The focus is represented as (-a, 0), and the directrix can be written as x=a.
2. The third equation is x2 = 4ay. In this equation, the parabola’s axis is y (positive axis), and the focus is represented as (0, a). The directrix of the parabola is written as y= -a.
3. The fourth equation is x2 = -4ay. The parabola’s axis is y (negative axis) in this equation. The parabola’s focus is represented as (0, -a), and the directrix is written as y=a.
   

The lengths of the latus rectum for all these four equations are 4a. These four equations are known as the standard equations of a parabola. These vary along with the two-dimension standard forms of the parabola. For the equations with y2, the axis of symmetry is the x-axis, and for the x2 equations, the axis of symmetry is the y-axis. 

Properties of a parabola

Here are some of the critical properties of a two-dimension standard form of a parabola.

• Conic: A parabola is a conic that is defined by its focal properties. The eccentricity of the parabola’s conic section determines its spherical properties — the greater the eccentricity, the lesser the spherical behaviour. And less eccentricity translates into more spherical behaviour. The eccentricity of the parabola is defined as the ratio of the distance between the focus and a point on the plane to the vertex.
• Symmetric: The parabola is symmetric on either side of its axis. It mainly displays symmetric properties concerning the x-axis.
• Focus and vertex connectivity: The parabola’s focus and vertex are connected via the axis. These points pass through the axis, connecting the focus and the vertex.
• Tangent: A tangent is a line touching the parabola. The tangent is parallel to the directrix at the vertices.

Conclusion

Understanding the concept of parabola becomes essential as the application of parabola extends to real-life examples. Parabolas are used in various real-life situations. Thus, it is crucial to understand the two-dimension standard forms of parabola, the equations related to the same, and the properties of a parabola that help solve numerous mathematical problems.