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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.
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.
x^{2} + y^{2} = r^{2}
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)
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
X^{2} + y^{2} = r^{2}
X = r sin t
Y = r cos t
(5 sin t)^{2} + (5 cos t)^{2} = r^{2}
5^{2} (sin^{2} t + cos^{2} t) = r^{2}
25 (sin^{2} t + cos^{2} t) = r^{2} Hence (sin^{2} t +
cos^{2} t = 1)
r^{2} = 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
X^{2} + y^{2} = r^{2}
X = r sin t
Y = r cos t
(2 sin t)^{2} + (2 cos t)^{2} = r^{2}
2^{2} (sin^{2} t + cos^{2} t) = r^{2}
4 (sin^{2} t + cos^{2} t) = r^{2} Hence (sin^{2} t +
cos^{2} t = 1)
r^{2} = 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`