Equation of the Circle in Various Forms

Equation of the Circle in Various Forms:

1. Standard Form:

  • Equation: (xh)2+(yk)2=r2
  • Center and Radius: Center at (h,k) and radius r.
  • Explanation: Represents a circle with center (h,k) and radius r.
  • Expanded Form:
    • x22hx+h2+y22ky+k2=r2
    • x2+y2+2gx+2fy+c=0, where g=h, f=k, and c=h2+k2r2.

2. General Form:

  • Equation: x2+y2+2gx+2fy+c=0
  • Parameters: g,f, and c are constants.
  • Center and Radius: Center at (g,f) and radius r=g2+f2c.
  • Explanation: Represents a circle centered at (g,f) and its radius can be deduced from the coefficients.
  • General Equation: Ax2+Ay2+Dx+Ey+F=0
    • Conditions: The coefficients A and B should be equal for the equation to represent a circle.
    • Center and Radius: The center of the circle is given by (D2A,E2A), and the radius is D2+E24AF4A2.

3. Parametric Form:

  • Equation: x=h+rcosθ and y=k+rsinθ
  • Parameters: h,k are the coordinates of the center, r is the radius, and θ varies from 0 to 2π.
  • Explanation: Represents the circle's points by varying the parameter θ through trigonometric functions.

4. Diameter Form:

  • Equation: (x1x2)2+(y1y2)2=(2r)2
  • Parameters: (x1,y1) and (x2,y2) are the endpoints of the diameter, and r is the radius.
  • Explanation: Represents a circle using the endpoints of its diameter and the distance formula.
  • Equation with Diameter Endpoints: If the endpoints of the diameter are (x1,y1) and (x2,y2), the equation of the circle is (xx1)(xx2)+(yy1)(yy2)=0.

5. Vector Form:

  • Equation: r=c+ru
  • Parameters: r is the position vector of any point on the circle, c is the position vector of the center, r is the radius, and u is the unit vector representing direction.
  • Explanation: Represents the circle using vectors, where the position vector r moves along the circle using the center c and the unit vector u.
  • Vector Equation: For a circle with center c and radius r, the vector equation is rc2=r2 where r is any point on the circle.

6. Implicit Form:

  • Equation: Ax2+Ay2+Dx+Ey+F=0
  • Parameters: A,D,E, and F are coefficients.
  • Explanation: Represents a circle in its implicit form, usually when the equation is given without directly identifying the center and radius.

7. Polar Form:

  • Equation: r=2acosθ
  • Parameters: r is the distance from the origin to any point on the circle, a is a constant radius, and θ varies from 0 to 2π.
  • Explanation: Represents the circle using polar coordinates, relating the radius r and the angle θ with a constant a.
  • Polar Equation:
    • r=2acos(θα)
    • r=2bsin(θβ) where a,b are the lengths of semi-major and semi-minor axes, and α,β are angles with respect to the coordinate axes.