General Equation of the Circle

General Equation of the Circle:

1. General Form of the Circle Equation:

  • The general equation of a circle is Ax2+Ay2+Dx+Ey+F=0.
  • Conditions: For this equation to represent a circle, A=B (coefficients of x2 and y2 terms should be equal).

2. Center and Radius from the General Equation:

  • Center: The center of the circle is given by (D2A,E2A).
  • Radius: r=D2+E24AF4A2.

3. Example of General Equation of Circle:

Given the equation 3x2+3y24x+6y5=0, determine its center and radius.

Solution:

Comparing the equation Ax2+Ay2+Dx+Ey+F=0 with 3x2+3y24x+6y5=0:

  • A=3, D=4, E=6, and F=5.

Using the formulas:

  • Center: (D2A,E2A)=(423,623)=(23,1)
  • Radius: r=(4)2+(6)243(5)432=16+36+6036=11236=3.111.76Therefore, the center of the circle represented by the equation 3x2+3y24x+6y5=0 is (23,1), and its radius is approximately 1.76.