Parametric Equation of a Circle

Parametric Equation of a Circle:

1. Definition:

  • Parametric equations describe the x and y coordinates of a point on a curve using a third parameter variable.
  • For a circle, parametric equations define x and y in terms of an angle or parameter t.

2. Parametric Equations for a Circle:

  • General Form:
    • x=h+rcos(t)
    • y=k+rsin(t)
  • h,k represent the center of the circle, r is the radius, and t is the parameter or angle.

3. Explanation:

  • x Coordinate: h+rcos(t) represents the x-coordinate of a point on the circle with center (h,k) and radius r at angle t.
  • y Coordinate: k+rsin(t) represents the y-coordinate of the same point on the circle.

4. Parameter t and Angle:

  • t usually varies from 0 to 2π for one complete revolution around the circle.
  • t is akin to the angle measured from the positive x-axis in the counterclockwise direction.

5. Example:

  • Circle with Center (3,2) and Radius 4:
    • x=3+4cos(t)
    • y=2+4sin(t)
    • 0t2π for a full revolution.

6. Interpretation:

  • For each t value in the range, the parametric equations generate corresponding x and y coordinates that trace the circumference of the circle with center (3,2) and radius 4.
  • As t varies from 0 to 2π, the point on the circle moves counterclockwise around its circumference, completing one revolution.