Circumference reduced equation

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?

The circle is made up of the set of all points equally distant from a fixed point on that plane.

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:

d_BECAUSE = 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?

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

Source: Brazil School - https://brasilescola.uol.com.br/matematica/equacao-reduzida-circunferencia.htm