Problem 77 - Precalculus 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 collection of points equidistant from a single point, the center. The equation of a circle can be derived from its definition, resulting in: (X-a)^2 + (Y-b)^2 = R^2, where (a,b) is the center of the circle, and R is the radius.

Step 1: Define a Circle

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

Step 2: Explain the Distance From the Center to Any Point on the Circle

The fixed distance from the center of the circle to any point on the circle is called the radius. Every point on the circle's circumference is exactly the same distance (the radius) from the circle's center.

Step 3: Deriving the Circle's Equation

The definition of the circle gives us a way to formulate the equation of a circle. Imagine a circle centered at origin. Any point (X,Y) on the circle meets the condition that it's a distance 'R' away from the center (0,0). The distance between two points is determined by the Pythagorean Theorem in the Cartesian plane. Here, since the distance is 'R', we have 'R' equals to the square root of (X^2 + Y^2), which, when squared, results in the equation of a circle: X^2 + Y^2 = R^2. Now, If the center is not at the origin (0,0), but at (a, b), the expression of distance changes slightly, resulting in the alternate form of the circle's equation: (X-a)^2 + (Y-b)^2 = R^2.