Parametric form of Circle

                                  
x = r cos , y = r sin ,0 < 2 represent the circle x² +y² = r², where is called parameter and the point P (r cos , r sin ) is called the point " " on the circle x² +y² = r².

Parametric form of the circle (x -h)² +(y -k)² = r²

Every point P on the circle can be represented as
      x = h + r cos , y = k + r sin , 0 < 2
Thus, x = h + r cos , y = k + r sin , 0 < 2 , represent the circle (x -h)² +(y -k)² = r².
is called parameter and the point (h +r cos , k +r sin ) is called the point " " on this circle.

Illustrative Examples

Example

Find the parametric equations of the circle x² +y² = 5

Solution

The given circle is x² + y² = 5
We know that the parametric equations of the circle x² +y² = r² are
     x = r cos , y = r sin , 0 < 2
The given circle is comparable with x² +y² = r², here r = 5
Therefore, the parametric equations of the given circle x² +y² = 5 are
    x = 5cos , y = 5 sin , 0 < 2

Example

Find the cartesian equations of the curves x = p +c cos , y = q +c sin , where is parameter. Do these equations represent a circle? If so, find center and radius.

Solution

Given x = p +c cos , y = q + c sin
=>    x -p = c cos , y -q = c sin
To eliminate the parameter , on squaring and adding these equations, we get
     (x -p)² + (y -q)² = c² (cos² +sin² )
=> (x -p)² +(y -q)² = c²,
which represents a circle with center (p, q) and radius = | c |.

Exercise

1. Find the parametric equations of the following circles :
   (i) x² +y² = 13
   (ii) (x -2)² +(y +3)² = 36
   (iii) x² + y² +4 x - 6 y -12 = 0
   (iv) 2 x² +2 y² = 5 x +7 y +3
   (v) x² + y² -2 a x - 2 a y = 0
   (vi) x² + y² + p x +q y = 0
2. Find the cartesian equations of the following curves:
   (i) x = 2 cos , y = 2 sin
   (ii) x = 1 +5 cos , y = 2 +5 sin
   (iii) x = -3 + 7cos , y = 4 +7 sin ,
   where is parameter. Do these equations represent circles? If so, find center and radius.

Answers

1. (i) x = 13 cos , y = 13 sin , 0 < 2
    (ii) x = 2 +6 cos , y = -3 +6 sin , 0 < 2
    (iii) x = -2 +5 cos , y = 3 +5 sin , 0 < 2
    (iv) x = cos , y = sin , 0 < 2
     (v) x = a +| a | cos , y = a +| a | sin , 0 < 2
     (vi) x = -p/2 +(1/2) (p² +q²) cos , y = -p/2 + (1/2)(p² +q²) sin , 0 < 2
2. (i) x² +y² = 4; circle, (0, 0), 2
    (ii) (x -1)² +(y -2)² = 25; circle, (1, 2), 5
    (iii) (x +3)² +(y -4)² = 7; circle, (-3 , 4), 7