## About Parametric equation of circle"

Parametric equation of circle :

Consider a circle with radius r and center at the origin. Let P(x, y) be any point on the circle. Assume that OP makes an angle θ with the positive direction of x-axis. Draw the perpendicular PM to the x-axis.

x/r = cosθ,  y/r = sinθ

Here x and y are the co-ordinates of any point on the circle. Note that these two co-ordinates depend on θ. The value of r is fixed.

The equations x = r cosθ, y = r sinθ are called the parametric equations of the circle  x2 + y2 = r

Here ‘θ’ is called the parameter and 0 ≤ θ ≤ 2π

Now let us look into some example problems on finding parametric equation of circle.

Example 1 :

Find the parametric equations of the circle x2 + y2 = 16

Solution :

Here r2 = 16 ⇒ r = 4

The parametric equations of the circle

x2 + y2 = r2 in parameter θ are x = r cosθ, y = r sin θ

The parametric equations of the given circle x2 + y2 = 16 are

x = 4 cos θ, y = 4 sin θ and 0 ≤ θ ≤ 2π

Example 2 :

Find the cartesian equation of the circle whose parametric equations are x = 2 cos θ, y = 2 sin θ, 0 ≤ θ ≤ 2π

Solution :

To find the caretsian equation of the circle, eliminate the parameter ‘θ’ from the given equations,

cos θ = x/2 ; sin θ = y/2

cos2θ + sin2θ  =  1

(x/2)2 + (y/2)2  =  1

x2 + y2 = 4 is the required cartesian equation of the circle.

Example 3 :

Find the cartesian equation of the circle whose parametric equations are x = 1/4 cosθ, y = 1/4 sin θ and 0 ≤ θ ≤ 2π

Solution :

To find the caretsian equation of the circle, eliminate the parameter ‘θ’ from the given equations,

x = (1/4) cosθ ; y = (1/4) sin θ

cosθ = 4x, sinθ = 4y

cos2θ + sin2θ  =  1

(4x)2 + (4y)2  =  1

16x2 + 16y2  =  1

16x2 + 16y2  =  1 is the required cartesian equation of the circle.

Example 4 :

Find the parametric equation of the circle 4x2 + 4y2 = 9

Solution :

4x2 + 4y2 = 9

Divide the equation by 4

x2 + y2 = (9/4)

Here r2 = 9/4 ⇒ r = 3/2

The parametric equations of the circle x2 + y2 = r2 in parameter θ are x = r cosθ, y = r sin θ

The parametric equations of the given circle

x = (3/2) cos θ, y = (3/2) sin θ and 0 ≤ θ ≤ 2π

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