Finding The Radius Of Circle J The Radius Of Circle J Is : R

In the realm of coordinate geometry, circles hold a fundamental place. They are defined by their center and radius, which dictate their position and size on the Cartesian plane. This article delves into the properties of a circle, particularly focusing on how to determine the radius given the equation of the circle and a point lying on its circumference. We will dissect the equation $(x-1)^2 + (y+1)^2 = r^2$, which represents a circle denoted as circle J. We are given that the point D(0, 3) lies on this circle, and our objective is to find the length of the radius, r. Understanding the equation of a circle is crucial for solving this problem. The standard form equation of a circle with center (h, k) and radius r is given by $(x-h)^2 + (y-k)^2 = r^2$. This equation is derived from the Pythagorean theorem and represents the set of all points (x, y) that are a distance r away from the center (h, k). By recognizing this standard form, we can extract valuable information about the circle, such as the center's coordinates and the radius's length. In this case, by comparing the given equation $(x-1)^2 + (y+1)^2 = r^2$ with the standard form, we can identify the center of circle J as (1, -1). The radius, r, is the distance from the center of the circle to any point on its circumference. Since we know the coordinates of the center and a point D(0, 3) on the circle, we can use the distance formula to calculate the radius. The distance formula, derived from the Pythagorean theorem, states that the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Applying this formula to our problem, we can calculate the distance between the center (1, -1) and the point D(0, 3), which will give us the radius r.

The equation of a circle is a powerful tool that encapsulates the geometric properties of a circle in a concise algebraic form. The general equation of a circle in the Cartesian plane is given by $(x - h)^2 + (y - k)^2 = r^2$, where (h, k) represents the coordinates of the center of the circle and r denotes the radius. This equation is derived from the Pythagorean theorem and embodies the fundamental definition of a circle: the set of all points equidistant from a central point. To effectively solve problems involving circles, it is imperative to understand how to extract information from this equation. The center coordinates (h, k) and the radius r are the key parameters that define a circle's position and size. By recognizing the standard form of the equation, we can readily identify these parameters and utilize them to solve various problems related to circles. Now, let's focus on the given equation: $(x - 1)^2 + (y + 1)^2 = r^2$. This equation represents circle J, and our goal is to determine the coordinates of its center. Comparing this equation with the general form, we can observe a direct correspondence between the terms. The term $(x - 1)^2$ corresponds to $(x - h)^2$, and the term $(y + 1)^2$ corresponds to $(y - k)^2$. This allows us to identify the values of h and k. We can see that h = 1, as the x-coordinate of the center is the value subtracted from x within the parentheses. Similarly, we can deduce that k = -1, as the y-coordinate of the center is the negative of the value added to y within the parentheses. Note that the equation has $(y + 1)^2$, which can be rewritten as $(y - (-1))^2$. Therefore, the center of circle J is located at the point (1, -1). This means that the circle is centered one unit to the right of the y-axis and one unit below the x-axis in the Cartesian plane. Knowing the center's coordinates is a crucial step in determining the radius of the circle, as the radius is the distance from the center to any point on the circle's circumference. In the next section, we will use the given point D(0, 3) and the center (1, -1) to calculate the radius.

Having determined the center of circle J to be (1, -1), we now proceed to calculate the radius, r. We are given that point D(0, 3) lies on the circle. The radius is defined as the distance from the center of the circle to any point on its circumference. Therefore, the distance between the center (1, -1) and the point D(0, 3) will give us the length of the radius. To calculate this distance, we employ the distance formula, a direct application of the Pythagorean theorem in coordinate geometry. The distance formula states that the distance d between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. This formula calculates the length of the hypotenuse of a right triangle formed by the horizontal and vertical distances between the two points. Applying the distance formula to our problem, we have the center (1, -1) as $(x_1, y_1)$ and the point D(0, 3) as $(x_2, y_2)$. Substituting these values into the formula, we get: $r = \sqrt{(0 - 1)^2 + (3 - (-1))^2}$. This equation represents the distance between the center of the circle and the point D. Now, let's simplify the equation step-by-step: First, we calculate the differences within the parentheses: $r = \sqrt{(-1)^2 + (4)^2}$. Next, we square the terms: $r = \sqrt{1 + 16}$. Finally, we add the squared terms and take the square root: $r = \sqrt{17}$. Therefore, the radius of circle J is $\sqrt{17}$ units. This means that every point on the circumference of circle J is a distance of $\sqrt{17}$ units away from the center (1, -1). The radius is a crucial parameter that defines the size of the circle, and we have successfully calculated it using the distance formula and the given information. In the following section, we will discuss the implications of this result and verify our answer against the provided options.

We have successfully determined the radius r of circle J to be $\sqrt{17}$. To ensure the accuracy of our solution, let's revisit the problem and the steps we took to arrive at the answer. We started with the equation of the circle, $(x - 1)^2 + (y + 1)^2 = r^2$, and identified the center as (1, -1). We then used the given point D(0, 3) lying on the circle and the distance formula to calculate the radius. The distance formula yielded $r = \sqrt{(0 - 1)^2 + (3 - (-1))^2} = \sqrt{17}$. Now, let's compare our result with the given options:

A. 17 B. 5 C. $\sqrt{17}$ D. $\sqrt{5}$

Our calculated radius, $\sqrt{17}$, matches option C. Therefore, the correct answer is C. This confirms that our calculations and reasoning are accurate. We have successfully found the radius of circle J using the equation of the circle, the coordinates of a point on the circle, and the distance formula. In conclusion, the radius r of circle J is $\sqrt{17}$ units. This problem demonstrates the application of fundamental concepts in coordinate geometry, such as the equation of a circle and the distance formula. By understanding these concepts and applying them systematically, we can solve a wide range of problems involving circles and other geometric figures. The ability to determine the radius of a circle given its equation and a point on its circumference is a valuable skill in mathematics and has applications in various fields, including physics, engineering, and computer graphics. This exercise highlights the importance of a strong foundation in mathematical principles and the ability to apply them effectively to solve real-world problems. Furthermore, it emphasizes the significance of verifying solutions to ensure accuracy and confidence in the results.