Finding The Radius Of Circle J Solving (x-1)^2 + (y+1)^2 = R^2

Introduction: Deciphering Circle J's Equation

In the fascinating realm of mathematics, circles hold a special place. Their elegant symmetry and fundamental properties have captivated mathematicians for centuries. One of the most powerful tools for understanding circles is their equation, a concise representation that encapsulates their essential characteristics. In this article, we embark on a journey to unravel the mysteries of a particular circle, Circle J, defined by the equation (x-1)^2 + (y+1)^2 = r^2. Our mission is to determine the length of its radius, denoted by r, given that the point D(0, 3) lies on the circle. This seemingly simple problem will lead us through a series of fundamental concepts in coordinate geometry, revealing the power of equations to describe geometric shapes.

The equation (x-1)^2 + (y+1)^2 = r^2 is a standard form representation of a circle in the Cartesian plane. This form, also known as the center-radius form, provides immediate insights into the circle's key attributes: its center and radius. The equation's structure directly corresponds to the distance formula, a cornerstone of coordinate geometry. It states that the distance between any point (x, y) on the circle and the circle's center (h, k) is equal to the radius r. In our case, the equation reveals that the center of Circle J is located at the point (1, -1). This is because the terms (x-1) and (y+1) within the equation directly indicate the horizontal and vertical shifts from the origin, respectively. The value on the right side of the equation, r^2, represents the square of the radius, a crucial piece of information that we will use to determine the actual radius.

Our task is to find the value of r, the radius of Circle J. We are given that the point D(0, 3) lies on the circle. This seemingly simple piece of information is the key to unlocking the value of r. Since point D lies on the circle, its coordinates must satisfy the circle's equation. This means that when we substitute x = 0 and y = 3 into the equation (x-1)^2 + (y+1)^2 = r^2, the equation must hold true. This substitution will transform the equation into an algebraic expression involving only one unknown, r^2, which we can then solve to find the value of r. This process exemplifies the power of coordinate geometry, where geometric relationships are translated into algebraic equations, allowing us to use the tools of algebra to solve geometric problems. The interplay between geometry and algebra is a recurring theme in mathematics, and this problem provides a tangible example of this connection.

Applying the Point-Circle Relationship: Finding r

To determine the radius r of Circle J, we will leverage the crucial information provided: the point D(0, 3) lies on the circle. As mentioned earlier, this implies that the coordinates of point D must satisfy the equation of Circle J. Therefore, we substitute x = 0 and y = 3 into the equation (x-1)^2 + (y+1)^2 = r^2. This substitution transforms the equation into a numerical expression that we can then simplify to isolate r^2.

Substituting x = 0, we get (0 - 1)^2, which simplifies to (-1)^2 = 1. This term represents the squared horizontal distance between point D and the center of the circle. Similarly, substituting y = 3, we get (3 + 1)^2, which simplifies to (4)^2 = 16. This term represents the squared vertical distance between point D and the center of the circle. Combining these results, our equation becomes 1 + 16 = r^2. This equation elegantly connects the coordinates of point D with the radius of Circle J. It states that the sum of the squared horizontal and vertical distances between point D and the center of the circle is equal to the square of the radius. This is a direct consequence of the Pythagorean theorem, which underlies the distance formula and the equation of a circle.

Simplifying the equation 1 + 16 = r^2, we obtain r^2 = 17. This equation tells us that the square of the radius of Circle J is 17. To find the actual radius r, we need to take the square root of both sides of the equation. Remember that the radius of a circle is a distance, and therefore must be a positive value. Taking the square root of 17, we get r = √17. This is the exact value of the radius of Circle J. It is an irrational number, meaning it cannot be expressed as a simple fraction, but it represents a precise length. Therefore, the radius of Circle J is √17 units. This result confirms that the correct answer choice is indeed C. √17.

Conclusion: Circle J and the Power of Equations

In this exploration, we have successfully determined the radius of Circle J, defined by the equation (x-1)^2 + (y+1)^2 = r^2, given that the point D(0, 3) lies on the circle. Our journey began with understanding the standard form equation of a circle and recognizing its connection to the distance formula and the Pythagorean theorem. We then applied the crucial piece of information that point D lies on the circle, allowing us to substitute its coordinates into the equation and transform it into an algebraic expression. By simplifying this expression, we isolated r^2 and subsequently found the value of r to be √17. This solution exemplifies the power of equations in describing geometric shapes and relationships.

This problem highlights the fundamental principles of coordinate geometry, where geometric concepts are translated into algebraic language. The equation of a circle, in its center-radius form, provides a concise and powerful representation of the circle's key attributes. The distance formula, a cornerstone of coordinate geometry, underlies the equation of a circle and allows us to calculate distances between points in the plane. The interplay between algebra and geometry is a recurring theme in mathematics, and this problem serves as a tangible example of this connection. By understanding these principles, we can effectively solve a wide range of geometric problems using algebraic techniques.

Furthermore, this exercise reinforces the importance of careful substitution and simplification in mathematical problem-solving. The accurate substitution of the coordinates of point D into the equation of Circle J was crucial for obtaining the correct result. Similarly, the step-by-step simplification of the resulting equation allowed us to isolate r^2 and determine the value of r. These skills are essential for success in mathematics and other quantitative disciplines. The ability to translate a problem into a mathematical form, apply appropriate techniques, and carefully interpret the results is a hallmark of mathematical proficiency. Therefore, this problem not only provides a solution to a specific question but also reinforces fundamental mathematical skills and concepts.

Answer

The radius r of circle J is √17, which corresponds to option C.

Repair Input Keyword

What is the value of r, the radius of circle J, given that the equation of circle J is (x-1)^2 + (y+1)^2 = r^2 and the point D(0,3) lies on the circle?

Title

Finding the Radius of Circle J Solving (x-1)^2 + (y+1)^2 = r^2