Equation of a Circle Given a Center and a Point: Detailed Guide

Understanding the equation of a circle is crucial for various applications in mathematics, engineering, and geometry. This article will guide you through the process of finding the equation of a circle when given its center and a point on the circle. We will explore the standard form of the circle's equation, the distance formula, and step-by-step instructions to solve the problem.

Standard Form of the Circle's Equation

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

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

Step-by-Step Solution: Finding the Equation of a Circle

1. Calculate the Radius

To find the radius (r), we use the distance formula between the center ((2, 3)) and the point ((-2, 5)) on the circle. The distance formula is:

( d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2} )

Substituting the given values:

( r sqrt{(-2 - 2)^2 (5 - 3)^2} sqrt{(-4)^2 2^2} sqrt{16 4} sqrt{20} 2sqrt{5} )

2. Substitute the Center and Radius into the Standard Form

Now that we have the radius, we can substitute the center ((2, 3)) and the radius (2sqrt{5}) into the standard form of the circle's equation:

[ (x - 2)^2 (y - 3)^2 (2sqrt{5})^2 ]

3. Simplify the Equation

Simplifying the right-hand side:

( (2sqrt{5})^2 4 cdot 5 20 )

Therefore, the equation of the circle is:

[ (x - 2)^2 (y - 3)^2 20 ]

Summary

We have derived the equation of the circle with the given center ((2, 3)) and passing through the point ((-2, 5)) using the steps of calculating the radius and substituting the values into the standard form of the circle's equation.

Practice similar problems and explore the broader applications of the circle's equation in real-world scenarios. This guide will help you master the concept of the circle's equation in standard form.

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