Problem 63 - College Algebra

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 points in a plane that are equidistant from a fixed point, which is the center. The fact that all points on the circle are of the same distance (which is the radius) away from the center leads to the equation of the circle. Although typically expressed with variables, conceptually, the equation insists that the squared distance from any point on the circle to the center equals to the square of the radius.

Step 1: Defining a Circle

A circle can be defined as a set of all points in a plane that are equidistant from a fixed point known as the center of the circle.

Step 2: Conceptualizing the Equation

To obtain the equation of the circle, we need to apply the concept of distance, considering the distance from any point on the circle to the center of the circle. This distance is a constant, which is referred to as the radius of the circle.

Step 3: Formulating the Equation

Keeping in mind that any point on the circle is at the same distance (the radius) from the center of the circle, we can thus say the squared distance from any point on the circle to the center is equal to the square of the radius. This idea is what constitute the standard form of a circle's equation, albeit usually expressed with variables.