General Equation of Circle

 What is General Equation of a Circle

If (h, k) is the co-ordinate of center of a circle and r is the radius of a circle, then Standard Equation of a Circle is:

(x – h)² + (y – k)² = r² ————————(1)

General Equation of a Circle is the alternate version of Standard Equation (1) of a Circle which is obtained by squaring the terms and simplifying.

As we know the formula

(a – b)² = a² + b² – 2ab

Apply above formula to Standard Equation of a Circle

(x – h)² + (y – k)² = r²

(x² + h² -2xh) + (y² + k² -2yk) = r²

Rearrage the terms

x² + y² – 2xh – 2yk + h² + k² – r² =0

x² + y² + 2x(-h) + 2y(-k) + h² + k² – r² =0

Substitute 

x = -g

y = -f

c = h² + k² – r²

we get

x² + y² + 2gx + 2fy + c = 0 ——————(2)

Equation (2) is named as the General Equation of a Circle. 

Here, 

(-g, -f) is the center of a circle and

 `sqrt{g^2+f^2-c}`  is the radius of a circle. 

If the General Equation of any Circle is given, we can find its Center and Radius by two methods: 

Click here for METHOD 1

Click here for METHOD 2

If the center and Radius of any circle is given, we can find its equation.

Click here for Solved Example 1

Click here for Solved Example 2

RELATED POST:

Definition of a Circle, Standard Equation of a Circle, Unit circle and its Equation

Solved EXAMPLE 1 on how to find Center and Radius of a circle from its Standard Equation

Solved EXAMPLE 2 on how to find Center and Radius of a circle from its Standard Equation