Finding The Center Of A Circle Given Its Equation (x+9)^2+(y-6)^2=10^2

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The fascinating world of geometry introduces us to various shapes and forms, each with its unique properties and equations. Among these, the circle holds a special place, admired for its symmetry and elegant simplicity. In this comprehensive guide, we'll delve deep into understanding the center of a circle, particularly when represented by an equation in the form (x - h)^2 + (y - k)^2 = r^2. This equation, known as the standard form or center-radius form, provides valuable insights into the circle's characteristics. We'll explore how to extract the coordinates of the circle's center directly from this equation and apply this knowledge to solve problems, ensuring a solid grasp of this fundamental concept in mathematics.

In the realm of analytical geometry, equations serve as powerful tools for describing geometric shapes. When we encounter an equation like (x + 9)^2 + (y - 6)^2 = 10^2, it represents a circle in the Cartesian plane. This form of the equation, known as the standard form or center-radius form, provides valuable information about the circle's properties. Specifically, it allows us to easily identify the coordinates of the circle's center and its radius. By carefully analyzing this equation, we can unlock key geometric insights and visualize the circle's position and size within the coordinate system. Let's embark on a journey to dissect this equation and extract the hidden information it holds.

The standard form of a circle's equation is given by:

(x - h)^2 + (y - k)^2 = r^2

Where:

  • (h, k) represents the coordinates of the center of the circle.
  • r represents the radius of the circle.

Now, let's analyze the given equation:

(x + 9)^2 + (y - 6)^2 = 10^2

To determine the center of the circle, we need to compare the given equation with the standard form. Notice that:

  • (x + 9)^2 can be rewritten as (x - (-9))^2.
  • (y - 6)^2 remains as is.
  • 10^2 represents the square of the radius, so the radius is 10.

By comparing the rewritten equation with the standard form, we can identify the values of h and k:

  • h = -9
  • k = 6

Therefore, the center of the circle is (-9, 6). This means that the circle is positioned in the Cartesian plane with its center located at the point where the x-coordinate is -9 and the y-coordinate is 6. The negative x-coordinate places the center to the left of the y-axis, while the positive y-coordinate positions it above the x-axis.

To find the center of the circle represented by the equation (x + 9)^2 + (y - 6)^2 = 10^2, we will follow a structured, step-by-step approach. This method ensures clarity and accuracy in identifying the center coordinates. By carefully comparing the given equation with the standard form of a circle's equation, we can systematically extract the required information. This process not only provides the solution but also reinforces our understanding of the relationship between a circle's equation and its geometric properties. Let's proceed with the solution, breaking it down into manageable steps for easy comprehension.

Here's how to find the center of the circle:

  1. Identify the Standard Form:

    The standard form of the equation of a circle is:

    (x - h)^2 + (y - k)^2 = r^2

    Where (h, k) is the center of the circle and r is the radius.

  2. Compare with the Given Equation:

    The given equation is:

    (x + 9)^2 + (y - 6)^2 = 10^2

    We need to rewrite the given equation to match the standard form. Notice that (x + 9) can be written as (x - (-9)). So, the equation becomes:

    (x - (-9))^2 + (y - 6)^2 = 10^2

  3. Determine the Center Coordinates:

    By comparing the rewritten equation with the standard form, we can identify the values of h and k:

    • h = -9
    • k = 6

    Therefore, the center of the circle is (-9, 6).

When presented with multiple-choice questions, a strategic approach to analyzing the options is crucial. In this case, we have four options for the center of the circle: A) (-9, 6), B) (-6, 9), C) (6, -9), and D) (9, -6). By carefully comparing these options with our calculated center coordinates, we can confidently identify the correct answer. This process not only validates our solution but also reinforces our understanding of the concept. Let's examine each option in detail and pinpoint the one that accurately represents the center of the circle.

Now, let's compare our result with the given options:

A. (-9, 6): This matches our calculated center. B. (-6, 9): This is incorrect. C. (6, -9): This is incorrect. D. (9, -6): This is incorrect.

Therefore, the correct answer is A. (-9, 6).

To truly master the concept of circle equations, it's essential to understand the underlying principles and key concepts. This knowledge not only enables us to solve problems effectively but also provides a deeper appreciation for the elegance and beauty of mathematics. In this section, we'll delve into the fundamental concepts related to circle equations, including the standard form, the general form, and the relationship between the equation and the circle's geometric properties. By grasping these core ideas, we can confidently tackle a wide range of problems involving circles and their equations.

Understanding circle equations involves several key concepts:

  • Standard Form of a Circle Equation: As discussed earlier, the standard form is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
  • Center-Radius Form: The standard form is also known as the center-radius form because it directly reveals the center and radius of the circle.
  • General Form of a Circle Equation: The general form is Ax^2 + Ay^2 + Bx + Cy + D = 0, where A, B, C, and D are constants. This form can be converted to the standard form by completing the square.
  • Relationship between Equation and Graph: The equation of a circle uniquely defines its position and size in the coordinate plane. By analyzing the equation, we can visualize the circle's center and radius, and vice versa.
  • Applications of Circle Equations: Circle equations have numerous applications in various fields, including geometry, physics, engineering, and computer graphics. They are used to model circular objects, analyze circular motion, and design circular structures.

In this comprehensive guide, we've embarked on a journey to unravel the mysteries of circle equations and master the art of identifying a circle's center. By understanding the standard form of the equation, (x - h)^2 + (y - k)^2 = r^2, we've learned how to directly extract the coordinates of the center, (h, k). We've applied this knowledge to solve a specific problem, confidently determining the center of a circle given its equation. Moreover, we've explored the key concepts underlying circle equations, including the relationship between the equation and the circle's geometric properties. With this newfound understanding, you're well-equipped to tackle a wide range of problems involving circles and their equations, solidifying your grasp of this fundamental concept in mathematics.

By understanding the standard form of the circle equation, we can easily identify the center and radius of a circle. This knowledge is crucial for solving various problems in geometry and related fields. The ability to interpret and manipulate circle equations is a valuable skill for any mathematics student or enthusiast.