Parametric Equations of a Parabola

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

• Concept of Parabola
• Standard Equation of a Parabola
• Standard form of Parabola y22 = - 4ax
• Standard form of Parabola x22 = 4ay
• Standard form of Parabola x22 = -4ay
• Parabola whose Vertex at a given Point and Axis is Parallel to x-axis
• Parabola whose Vertex at a given Point and Axis is Parallel to y-axis
• Position of a Point with respect to a Parabola
  
• Parametric Equations of a Parabola
• Parabola Formulae
• Problems on Parabola

11 and 12 Grade Math 

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