# 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