Finding The Radius Of A Circle Given Its Equation (x+5)^2+(y-3)^2=4^2

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Decoding the Circle Equation

When confronted with the equation of a circle in the form (x-h)^2 + (y-k)^2 = r^2, we're presented with a powerful tool to understand the circle's characteristics. This standard form equation holds the key to unlocking the circle's center coordinates and, most importantly for our query, its radius. Let's delve into the anatomy of this equation and how it relates to the circle's properties.

At its core, the equation represents all the points (x, y) that lie on the circumference of the circle. The values 'h' and 'k' pinpoint the circle's center coordinates in the Cartesian plane, while 'r' signifies the circle's radius – the distance from the center to any point on the circle. It's crucial to recognize that the 'r' in the equation is squared, meaning the radius is the square root of the constant term on the right-hand side of the equation. Understanding this fundamental relationship is paramount to accurately determining the radius when presented with a circle's equation.

In the given equation, (x+5)^2 + (y-3)^2 = 4^2, we can directly extract the values that define our circle. By comparing this equation to the standard form, we observe that 'h' corresponds to -5 (note the sign change due to the (x - h) format), 'k' corresponds to 3, and the constant term, which equals r^2, is 4^2 or 16. This immediately tells us that the square of the radius is 16. To find the radius itself, we simply take the square root of 16, which yields 4. This process underscores the elegance and efficiency of the standard form equation in revealing a circle's essential features.

Therefore, by carefully dissecting the equation and recognizing the roles of 'h', 'k', and 'r', we've successfully determined that the radius of the circle represented by the equation (x+5)^2 + (y-3)^2 = 4^2 is 4 units. This showcases how a solid grasp of the standard form equation empowers us to quickly and accurately extract critical information about circles.

Identifying the Radius

To accurately identify the radius of the circle represented by the equation (x+5)^2 + (y-3)^2 = 4^2, we must first understand the general form equation of a circle. This general equation, as we discussed, 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 represents the radius. The critical step in determining the radius lies in correctly interpreting the equation provided and mapping its components to the general form.

In our given equation, (x+5)^2 + (y-3)^2 = 4^2, we can observe a direct correspondence with the general form. The term (x + 5)^2 can be rewritten as (x - (-5))^2, highlighting that the x-coordinate of the center, 'h', is -5. Similarly, the term (y - 3)^2 directly indicates that the y-coordinate of the center, 'k', is 3. These observations pinpoint the center of the circle at the coordinates (-5, 3). However, our primary focus is the radius, which is determined by the term on the right side of the equation.

The right side of our equation is 4^2, which equals 16. This value represents r^2, the square of the radius. To find the radius, 'r', we need to take the square root of 16. The square root of 16 is 4. Therefore, the radius of the circle described by the equation (x+5)^2 + (y-3)^2 = 4^2 is 4 units. This straightforward calculation demonstrates how a clear understanding of the circle's equation allows for easy extraction of its radius.

This process underscores the importance of recognizing the relationship between the equation's components and the circle's properties. By systematically comparing the given equation to the general form and applying the square root operation, we confidently arrive at the solution: the radius of the circle is 4 units. This analytical approach is crucial for solving similar problems in geometry and beyond.

Solution and Explanation

The question at hand asks for the radius of a circle defined by the equation (x+5)^2 + (y-3)^2 = 4^2. As we've established, the key to solving this problem lies in recognizing the standard form of a circle's equation: (x - h)^2 + (y - k)^2 = r^2. In this equation, (h, k) represents the center of the circle, and r represents its radius. Our task is to extract the value of 'r' from the given equation.

By comparing the given equation, (x+5)^2 + (y-3)^2 = 4^2, with the standard form, we can identify the corresponding components. The term (x + 5)^2 can be rewritten as (x - (-5))^2, indicating that h = -5. The term (y - 3)^2 directly shows that k = 3. These values define the center of the circle as (-5, 3). However, our primary concern is the radius, which is determined by the right-hand side of the equation.

The right-hand side of the equation is 4^2, which equals 16. This value corresponds to r^2, the square of the radius. To find the radius 'r', we need to take the square root of 16. The square root of 16 is 4. Therefore, the radius of the circle is 4 units. This directly corresponds to option B in the given choices.

This solution highlights the power of recognizing and applying the standard form equation of a circle. By directly comparing the given equation to the standard form, we can quickly and accurately identify the circle's center and radius. The process involves rewriting terms if necessary to match the (x - h) and (y - k) format and then taking the square root of the constant term on the right side to find the radius. This method provides a clear and efficient way to solve problems involving circles and their equations.

Therefore, the correct answer is B. 4 units. We've demonstrated how understanding the equation of a circle and applying basic algebraic principles leads us to the accurate solution.

Answer

B. 4 units