Introduction

In this article, we will delve into the process of deriving the equation of a circle given its center and radius. Specifically, we will find the equation of the circle with center at (2, -3) and a radius of 4. We will closely follow the general form of the equation of a circle and provide detailed steps to reach the final equation.

The General Form of a Circle's Equation

The equation of a circle with center ((h, k)) and radius (r) is given by:

[ (x - h)^2 (y - k)^2 r^2 ]

This is the standard form of the equation of a circle, which directly relates the coordinates of any point ((x, y)) on the circle to the circle's center and radius. Here, (h) and (k) represent the center coordinates, and (r) is the radius.

Deriving the Equation for a Circle with Center (2, -3) and Radius 4

Given:

Center: ((2, -3)) Radius: 4

Substitute (h 2), (k -3), and (r 4) into the standard form of the circle's equation:

[ (x - 2)^2 (y - (-3))^2 4^2 ]

Simplifying the equation, we get:

[ (x - 2)^2 (y 3)^2 16 ]

This is the equation of the circle with the given center and radius.

Expanding the Equation

To fully expand and standardize the equation, let's expand the terms:

[ (x - 2)^2 (y 3)^2 16 ]

[ x^2 - 4x 4 y^2 6y 9 16 ]

[ x^2 y^2 - 4x 6y 13 16 ]

[ x^2 y^2 - 4x 6y - 3 0 ]

So, the expanded form of the equation of the circle is:

[ x^2 y^2 - 4x 6y - 3 0 ]

Summary

In summary, the equation of the circle with center (2, -3) and radius 4 is:

[ (x - 2)^2 (y 3)^2 16 ]

Additional Information

The same process can be applied to other circles by simply substituting the respective center coordinates and radius. If you need further assistance or more examples, feel free to check other related articles or textbooks on coordinate geometry.