Parametric Coordinates of Parabola

A parabola is a plain cover and an important member of the conical sections’ family. A parabola can be defined with the help of an equation. Parametric equations are used to express coordinates of the point that make up a geometrical object, such as a parabola.

The parameter of a parabola can be determined with the help of the parametric equation of a parabola. Such a parametric equation also exists for an ellipse. Any parameter ki equations can be easily graphed on the desmos graphing calculator. It is a tool that can be found online.

Parametric Equations

A parametric equation defines a group of quantities as functions of singular or plural independent variables called the parameters. They express the coordinates of a group of points that make up a geometric object, such as s parabola. 

The parametric representations are not unique in general. The use of parametric equations is usually done in kinematics. Let us find the parametric equations of various geometric objects.

The Parameter of a Parabola

The parameter of a parabola can be easily defined with the help of the parametric equation of the parabola. The best form to represent the parametric coordinates of any point on the parabola with the standard equation y² = 4ax is (at², 2at).

And for all the values of t, the coordinates

(at², 2at ). All the values of t will satisfy the equation of the parabola that is y² = 4ax.

Now let us discuss the parameter of a parabola and the parametric coordinates of the parabola.

As given, the standard equation of the parabola is y² = – 4ax.

So, the parametric coordinates of the parabola with the standard equation y² = -4ax 

will be (-at², 2at).

Hence, the parameter of a parabola in the form of coordinates will be:

x = -at² and y = 2at.

Now, if the standard equation of the parabola is x² = 4ay, 

then the parameter coordinates of the parabola will be ( 2at, -at² ).

So, the parameter of a parabola in the form of an equation will be x² = 4ay,

where x = 2at and y= at²

Now, consider a parabola with the standard equation x² = -4ay.

The parametric coordinates of this parabola will be (2at, – at²).

So, the parameter ki equation of this parabola will be x² = -4ay, where x = 2at and y = -at².

If the standard equation of the parabola will be (y – k)² = 4a(x – h)2, then the parametric equation will be x = h + at² and y = k + 2at.

Parametric Equation of an Ellipse

An ellipse is a geometrical shape that is a locus of all points which satisfy the given condition that is x = a × cost and y = b × sint.

The parametric equation of an ellipse is quite similar to the parametric equation of a circle.

If the center of an ellipse is on the (0, 0), then the parametric equations of the ellipse will be

x = a × cost and y = b × sint.

Here, a is the radius of the ellipse along the x-axis, and b is the radius of the ellipse along the Y-axis.

For the ellipse, which is centered at the origin, we add offsets to x and y terms to move the ellipse to the correct location.

Now, the parametric equations for ellipse which is not centered at the origin will become:

c = h + a cost and y = k + b sint.

Here the (h, k) is the x,y coordinates of the center of the ellipse.

The parametric equations are only true for ellipses that are aligned with coordinate planes, that is, the ellipses in which the major and minor axes are parallel to the coordinates.

Parametric Equations on Desmos

Desmos is a graphing calculator that is available as a web application and mobile application.

We can easily craft the parametric equations on the small graphic calculator. It is as easy as plotting an ordered pair. Instead of using the numerical coordinates, use the expression in terms of t, for example, cos d and sin t. With Desmos, the curves, lines, and relations can be graphed with ease.

Conclusion

The geometric shapes such as parabolas and ellipses can be defined in terms of parametric equations with the help of parametric coordinates. The parameter of a parabola with the standard equation  y² = 4ax will have parametric coordinates as x= at² and y = 2at. Together, they have termed the parametric equations of the parabola.

The parameter ki equations for ellipse defer based on whether the ellipse is centered at the origin or not centered at the origin. Desmos is an advanced graphic calculator which is available in the form of web applications and mobile applications.