Equation of a Circle Formula

Circle equation formula refers to the equation of a circle which represents the center-radius form of the circle. To recall, circle is referred to a round shape boundary where all the points are equidistant from the centre. An equation is generally required to represent the circle. There are basically two forms of representation:

  1. Standard Form
  2. General Form

Standard Form of Circle Equation

(x – a)2 + (y – b)2 = r2

Where,

  • a, b is the center,
  • r is the radius

General form of Circle Equation

x2 + y2 + Ax + By + C = 0

Solved Examples

Question 1: If the center point and radius of a circle is given as (4, 5) and 7 respectively. Represent this as a circle equation ?

Solution:

Given parameters are

Center (a, b) = (4, 5);

radius r = 7

The standard form of circle equation is,

(x − a)2 + (y − b)2 = r2

So, (x − 4)2 + (y − 5)2 = 72

So, (x − 4)2 + (y − 5)2 = 49

Question 2: Find the centre and radius of the circle whose equation is given by x2 + y2 – 10x + 14y + 38 = 0.
Solution:
Given circle equation is:
x2 + y2 – 10x + 14y + 38 = 0
x2 -2(5)x + y2 + 2(7)y + 38 = 0
This can also be written as:
x2 -2(5)x + (5)2 + y2 + 2(7)y + (7)2 + 38 – 25 – 49 = 0
(x – 5)2 + (y + 7)2 – 36 = 0
(x – 5)2 + (y + 7)2 = 36
(x – 5)2 + (y + 7)2 = 62
Comparing this with the standard form (x – a)2 + (y – b)2 = r2
a = 5, b = -7 and r = 6
Therefore, centre = (5, -7) and radius = 6 units.

Question 3: Write the general form of the circle equation with centre (2, 3) and radius 1 unit.
Solution:
Given,
Centre = (a, b) = (2, 3)
Radius = r = 1
Standard form of circle equation is (x – a)2 + (y – b)2 = r2
Substituting the values of centre and radius,
(x – 2)2 + (y – 3)2 = 12
x2 – 4x + 4 + y2 – 6y + 9 = 1
x2 + y2 – 4x – 6y + 10 – 1 = 0
x2 + y2 – 4x – 6y + 9 = 0
This of the form x2 + y2 + Ax + By + C = 0 where A = -4, B = -6, C = 9
Hence, the general form of the circle equation is x2 + y2 – 4x – 6y + 9 = 0

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