We will learn in the simplest way how to find the parametric equations of a parabola.
The best and easiest form to represent the co-ordinates of any point on the parabola y\(^{2}\) = 4ax is (at\(^{2}\), 2at). Since, for all the values of ‘t’ the coordinates (at\(^{2}\), 2at) satisfy the equation of the parabola y\(^{2}\) = 4ax.
Together the equations x = at\(^{2}\) and y = 2at (where t is the parameter) are called the parametric equations of the parabola y\(^{2}\) = 4ax.
Let us discuss the parametric coordinates of a point and their parametric equations on the other standard forms of the parabola.
The following gives the parametric coordinates of a point on four standard forms of the parabola and their parametric equations.
Standard equation of the parabola y\(^{2}\) = -4ax:
Parametric coordinates of the parabola y\(^{2}\) = -4ax are (-at\(^{2}\), 2at).
Parametric equations of the parabola y\(^{2}\) = -4ax are x = -at\(^{2}\), y = 2at.
Standard equation of the parabola x\(^{2}\) = 4ay:
Parametric coordinates of the parabola x\(^{2}\) = 4ay are (2at, at\(^{2}\)).
Parametric equations of the parabola x\(^{2}\) = 4ay are x = 2at, y = at\(^{2}\).
Standard equation of the parabola x\(^{2}\) = -4ay:
Parametric coordinates of the parabola x\(^{2}\) = -4ay are (2at, -at\(^{2}\)).
Parametric equations of the parabola x\(^{2}\) = -4ay are x = 2at, y = -at\(^{2}\).
Standard equation of the parabola (y - k)\(^{2}\) = 4a(x - h):
The parametric equations of the parabola (y - k)\(^{2}\) = 4a(x - h) are x = h + at\(^{2}\) and y = k + 2at.
Solved examples to find the parametric equations of a parabola:
1. Write the parametric equations of the parabola y\(^{2}\) = 12x.
Solution:
The given equation y\(^{2}\) = 12x is of the form of y\(^{2}\) = 4ax. On comparing the equation y\(^{2}\) = 12x with the equation y\(^{2}\) = 4ax we get, 4a = 12 ⇒ a = 3.
Therefore, the parametric equations of the given parabola are x = 3t\(^{2}\) and y = 6t.
2. Write the parametric equations of the parabola x\(^{2}\) = 8y.
Solution:
The given equation x\(^{2}\) = 8y is of the form of x\(^{2}\) = 4ay. On comparing the equation x\(^{2}\) = 8y with the equation x\(^{2}\) = 4ay we get, 4a = 8 ⇒ a = 2.
Therefore, the parametric equations of the given parabola are x = 4t and y = 2t\(^{2}\).
3. Write the parametric equations of the parabola (y - 2)\(^{2}\) = 8(x - 2).
Solution:
The given equation (y - 2)\(^{2}\) = 8(x - 2) is of the form of (y - k)\(^{2}\) = 4a(x - h). On comparing the equation (y - 2)\(^{2}\) = 8(x - 2) with the equation (y - k)\(^{2}\) = 4a(x - h) we get, 4a = 8 ⇒ a = 2 , h = 2 and k = 2.
Therefore, the parametric equations of the given parabola are x = 2t\(^{2}\) + 2 and y = 4t + 2.
● The Parabola
11 and 12 Grade Math
From Parametric Equations of a Parabola to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Dec 06, 23 01:23 AM
Dec 04, 23 02:14 PM
Dec 04, 23 01:50 PM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.