Reduced equation of circumference it has several applications in our daily lives, such as radar and tsunami detection. The circle has two elements: o center it's the lightning, which is the distance from the center to the edge of the circle.
Just like the straight, it is possible to determine the equation of a circle knowing the coordinates of the center and the measure of its radius. There is more than one way to represent a circle algebraically, however, we will emphasize the reduced equation of the circumference.
read more: Circumference elements: know what they are
How to determine the reduced equation of the circumference?
A circle is the set of points of the Cartesian plane that are equidistant from a given point, that is, from the center of the circumference. At this distance let's call it lightning, that is, we are going to “collect” points of the form P (x, y) that have the same distance from the center.
Consider a circle with center C (a, b) and radius r:
We are interested in the points that satisfy the condition that the distance between C and P is equal to lightning, i.e:
dBECAUSE = r
Gives distance between two points, we have:
Thus, the reduced equation of the circle that has center C (a, b) and radius r is given by:
Examples
- The equation (x – 3)2 + (y – 4)2 = 169 represents a circle with center C (3, 4) and radius r2 = 169, ie r = 13.
- the x equation2 + y2 = 0 represents a circle centered on the origin of the coordinate system and radius 0.
- The equation (x + 4)2 + (y – 4)2 = 169 also represents a circle with center C (-4, 4) and radius 13.
See too: How to find the center of a circle?
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solved exercises
Question 1 - (PUC-RS) According to FIFA rule 2, the official soccer ball must have its largest circumference measuring from 68 cm to 70 cm. Considering the 70 cm circumference and using a Cartesian referential to represent it, as in the following drawing, we could say that its equation is:
Solution:
We know that the circumference length is given by:
Since the circle has the center at the origin of the coordinate system, the coordinate of the center is C (0, 0). Now, replacing the information in the formula for the equation of the circle, we will have:
by Robson Luiz
Maths teacher
Would you like to reference this text in a school or academic work? Look:
LUIZ, Robson. "Reduced equation of the circumference"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/equacao-reduzida-circunferencia.htm. Accessed on June 28, 2021.
