Learn about parametric equations based on circle here and understand the concept better with solved examples provided. For more help, students can connect to an online tutor anytime and get the required help in the concept.

  • In mathematics, parametric equations are a method of defining a relation using parameters. A simple kinematical example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion.

  • Abstractly, a Parametric Equation defines a relation as a set of equations. It is therefore somewhat more accurately defined as a parametric representation. It is part of regular parametric representation.

Source: Wikipedia.

 

 

Parametric equation of a circle

 

A parametric equation of circle is the coordinates of a point on the circle in terms of a single variable θ. These single variables are called as parameter. The parametric equations of a circle with radius (r ≥ 0) and center (h, k) are given by

x = h + r cos θ          0   ≤ θ ≤ 2`pi`

y = k + r sin θ

Parametric equation of a circle centered at the origin with the radius r.

x2 + y2 = r2

Equation of a circle with single variable formula     

x = ± `sqrt(r^2 - y^2)`  

y = ± `sqrt(r^2 - x^2)` 

Each formula gives a portion of a circle

y= `sqrt(r^2 - x^2)`           (Top)

y= - `sqrt(r^2 - x^2)`        (Bottom)

x= `sqrt(r^2 - y^2)`           (Right side)

x = - `sqrt(r^2 - y^2)`          (Left side)              

 

Parametric equation Examples

 

Below are provided parametric equation examples for a better understanding:

Example 1:

Find the radius of parametric equation of the circle for the given x = 5 sin t and y = 5 cos t where 0 < t < `pi`

Solution:

Given

x= 5 sin t

y = 5 cos t

Formula for the parametric equation of a circle

X2 + y2 = r2

X = r sin t

Y = r cos t

(5 sin t)2 + (5 cos t)2 = r2

52 (sin2 t + cos2 t) = r2

25 (sin2 t + cos2 t) = r2                Hence (sin2 t + cos2 t = 1)

r2 = 25

Radius of the parametric circle is 5.

Example 2:

Find the radius of parametric equation of the circle for the given x = 2 sin t and y = 2 cos t where 0 < t < `pi`

Solution:

Given

x= 2 sin t

y = 2 cos t

Formula for the parametric equation of a circle

X2 + y2 = r2

X = r sin t

Y = r cos t

(2 sin t)2 + (2 cos t)2 = r2

22 (sin2 t + cos2 t) = r2

4 (sin2 t + cos2 t) = r2                Hence (sin2 t + cos2 t = 1)

r2 = 4

Radius of the parametric circle is 2.

 

Example 3:

Find the parametric equation of a circle with radius= 6 and center (2, 4).

Solution:

Given

(h, k) =(2, 4)

Radius = 6

Let

x = h + r cos `theta`

y = k + r sin `theta`

Parametric equation of a circle with radius 6 and center (2, 4)

x = 2 + 6 cos `theta`

y = 4 + 6 sin `theta`