**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 x^{2} + y^{2} = r^{2 }

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 x^{2} + y^{2} = 16

**Solution :**

Here r^{2} = 16 ⇒ r = 4

The parametric equations of the circle

x^{2} + y^{2} = r^{2} in parameter θ are x = r cosθ, y = r sin θ

The parametric equations of the given circle x^{2} + y^{2} = 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

cos^{2}θ + sin^{2}θ = 1

(x/2)^{2} + (y/2)^{2} = 1

x^{2} + y^{2} = 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

cos^{2}θ + sin^{2}θ = 1

(4x)^{2} + (4y)^{2} = 1

16x^{2} + 16y^{2} = 1

16x^{2} + 16y^{2} = 1 is the required cartesian equation of the circle.

**Example 4 :**

Find the parametric equation of the circle 4x^{2} + 4y^{2} = 9

**Solution :**

4x^{2} + 4y^{2} = 9

Divide the equation by 4

x^{2} + y^{2} = (9/4)

Here r^{2} = 9/4 ⇒ r = 3/2

The parametric equations of the circle x^{2} + y^{2} = r^{2} 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

- How to find the length of tangent
- How to find equation of tangent at the given point
- Inside outside or on the circle

After having gone through the stuff given above, we hope that the students would have understood "Parametric equation of circle".

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