PARAMETRIC EQUATION OF A CIRCLE WORKSHEET

(1)  Find the parametric equations of the circle x² + y² = 16

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

(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

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

Answers

Question 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

Question 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.

Question 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.

Question 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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