Problem 77 - 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 all points at an equal distance from a singular point (the center). Using this definition, an arbitrary point on the circle's distance to the center will always be equal to the radius. The square of this distance (to remove the square root in the distance formula) gives an equation resembling the standard form of the equation of the circle.

Step 1: Understanding A Circle

A circle is a geometric shape that is defined as a set of all points in a plane that are at a constant distance (sometimes called the radius) from a fixed point (often called the center)

Step 2: Applying The Definition

Take an arbitrary point on the circle. According to the definition of a circle, the distance between this point and the center of the circle is equal to the radius of the circle. This concept is integral to deriving an equation for a circle.

Step 3: Formulating The Equation

Based on the distance formula in geometry, the distance between two points can be given by the square root of the sum of the squares of the differences of their coordinates. In our situation, the two points are the center of the circle and the arbitrary point on the circle. Since the distance between these two points is the radius of the circle and it's always constant, we square both sides (to remove the square root) and get an equation that resembles the standard form of equation of a circle.