Problem 77 - College Algebra Essentials

Question

What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.

Step-by-Step Solution

Answer

A circle is defined as a set of all points in a plane that lie at the same distance, the radius, from a fixed point, the center. From this concept, the equation of a circle can be formed which states that for each point on the circle, the distance between that point and the center of the circle will always be equal to the radius.

Step 1: Understanding a circle

The first thing to understand about a circle is that it is made up of all points that are the same distance from a given point, which we call the center. This consistent distance is known as the radius of the circle.

Step 2: Relation between the center, radius and any point on the circle

Every point on the circle, due to the nature of a circle, lies at the same distance (radius) from the center. This is an important characteristic of the circle and is what actually defines it as a circle. Regardless of where a point on the circle is, it will always have this fixed distance to the center.

Step 3: Formulating the definition into an equation

From this definition, a form of an equation can be generated. This equation states that for each point on the circle, the distance between that point and the center of the circle will always be equal to the radius. In other words, for every point (X,Y) on the circle, the square of its distance from the center (a,b) equals the square of the radius. This forms the standard equation of a circle. This equation is represented algebraically when variables are introduced, but the concept remains the same.