The circle is a flat figure that can be represented in the Cartesian plane, using the studies related to Analytical Geometry, responsible for establishing relationships between algebra and geometry. The circle can be represented on the coordinate axis using an equation. One of these mathematical expressions is called the normal equation of the circle, which we will study next.
The normal equation of the circumference is the result of developing the reduced equation. Look:
(x – a) ² + (y – b) ² = R²
x² – 2ax + a² + y² – 2by + b² = R²
x² - 2ax + a² + y² - 2by + b² - R² = 0
x² + y² - 2ax - 2by + a² + b² - R² = 0
Let's determine the normal equation of the circle with center C (3, 9) and radius equal to 5.
(x – a) ² + (y – b) ² = R²
(x – 3)² + (y – 9)² = 5²
x² – 6x + 9 + y² – 18y + 81 – 25 = 0
x² + y² - 6x - 18y + 65 = 0
We can also use the expression x² + y² – 2ax – 2by + a² + b² – R² = 0, observe the development:
x² + y² – 2*3*x – 2*9*y + 3² + 9² – 5² = 0
x² + y² – 6x – 18y + 9 + 81 – 25 = 0
x² + y² - 6x - 18y + 65 = 0
From the normal equation of the circle we can establish the coordinates of the center and the radius. Let's perform a comparison between the equations x² + y² + 4x – 2y – 4 = 0 and x² + y² – 2ax – 2by + a² + b² – R² = 0. Note the calculations:
x² + y² + 4x – 2y – 4 = 0
x² + y² - 2ax - 2by + a² + b² - R² = 0
– 2a = 4 → a = – 2
– 2 = – 2b → b = 1
a² + b² - R² = - 4
(– 2)² + 12 – R² = – 4
4 + 1 - R² = - 4
– R² = – 4 – 4 – 1
– R² = – 9
R² = 9
√R² = √9
R = 3
Therefore, the normal equation of the circle x² + y² + 4x – 2y – 4 = 0 will have center C (-2, 1) and radius R = 3.
by Mark Noah
Graduated in Mathematics
Brazil School Team
Analytical Geometry - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/equacao-normal-circunferencia.htm