Circle Equation X²+y²+4x-6y-36=0 Analysis And Solutions
In the realm of analytical geometry, circles hold a position of fundamental importance. Understanding their equations and properties unlocks a gateway to solving a myriad of geometric problems. This article delves into the intricacies of a circle defined by the equation x² + y² + 4x - 6y - 36 = 0, providing a step-by-step guide to unravel its characteristics and explore related concepts.
Understanding the General Equation of a Circle
Before diving into the specifics of our given equation, let's first establish a solid understanding of the general equation of a circle. This equation serves as the foundation for analyzing any circle within the Cartesian coordinate system. The general form of a circle's equation is expressed as:
x² + y² + 2gx + 2fy + c = 0
Where:
- (-g, -f) represents the coordinates of the center of the circle.
- √(g² + f² - c) determines the radius of the circle.
This general form provides a powerful framework for extracting key information about a circle, such as its center and radius, which are crucial for various geometric calculations and constructions.
Transforming the General Equation to Standard Form
The standard form of a circle's equation offers a more intuitive representation, directly revealing the circle's center and radius. The standard form is given by:
(x - h)² + (y - k)² = r²
Where:
- (h, k) represents the coordinates of the center of the circle.
- r denotes the radius of the circle.
Converting the general equation to standard form involves a process known as completing the square. This technique allows us to rewrite quadratic expressions in a more manageable form, ultimately leading to the standard equation of the circle. Let's embark on this transformation for our given equation:
x² + y² + 4x - 6y - 36 = 0
Step-by-Step Conversion
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Group the x and y terms: Begin by rearranging the equation, grouping the x terms together and the y terms together: (x² + 4x) + (y² - 6y) = 36
This step sets the stage for completing the square for both the x and y variables.
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Complete the square for x: To complete the square for the x terms, we need to add and subtract the square of half the coefficient of the x term. In this case, the coefficient of x is 4, so half of it is 2, and the square of 2 is 4. Adding and subtracting 4 within the parentheses, we get: (x² + 4x + 4 - 4) + (y² - 6y) = 36
The expression x² + 4x + 4 now forms a perfect square trinomial.
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Complete the square for y: Similarly, for the y terms, the coefficient of y is -6, half of which is -3, and the square of -3 is 9. Adding and subtracting 9 within the parentheses, we obtain: (x² + 4x + 4 - 4) + (y² - 6y + 9 - 9) = 36
The expression y² - 6y + 9 now forms another perfect square trinomial.
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Rewrite as squared terms: Now, we can rewrite the perfect square trinomials as squared terms: (x + 2)² - 4 + (y - 3)² - 9 = 36
This step highlights the squared terms that define the circle's equation in standard form.
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Isolate the squared terms and constant: Move the constant terms to the right side of the equation: (x + 2)² + (y - 3)² = 36 + 4 + 9 (x + 2)² + (y - 3)² = 49
This equation is now in the standard form of a circle's equation.
Identifying the Center and Radius
By comparing the equation (x + 2)² + (y - 3)² = 49 with the standard form (x - h)² + (y - k)² = r², we can readily identify the center and radius of the circle:
- Center: The center of the circle is at the point (-2, 3).
- Radius: The radius of the circle is √49 = 7 units.
Analyzing the Statements
Now that we have successfully converted the equation to standard form and determined the center and radius, let's analyze the given statements:
A. To begin converting the equation to standard form, subtract 38 from both sides.
This statement is incorrect. As shown in our step-by-step conversion, the initial step involves grouping the x and y terms and moving the constant term to the right side of the equation. Subtracting 38 from both sides is not a necessary or helpful step in this process.
B. To complete the square for the x terms, add (4/2)² = 4 to both sides.
This statement is accurate. Completing the square for the x terms involves adding the square of half the coefficient of the x term, which is (4/2)² = 4, to both sides of the equation. This step allows us to rewrite the x terms as a perfect square trinomial.
Key Concepts in Circle Equations
Understanding circle equations involves grasping several key concepts:
- General Form: The general equation x² + y² + 2gx + 2fy + c = 0 provides a comprehensive representation of a circle.
- Standard Form: The standard equation (x - h)² + (y - k)² = r² directly reveals the center (h, k) and radius r of the circle.
- Completing the Square: This technique is crucial for converting the general equation to standard form, enabling easy identification of the circle's center and radius.
Applications of Circle Equations
The knowledge of circle equations extends far beyond theoretical exercises. It finds practical applications in various fields, including:
- Engineering: Designing circular structures, calculating stress distribution in circular components.
- Computer Graphics: Rendering circles and circular arcs in images and animations.
- Navigation: Determining distances and bearings using circular arcs.
- Astronomy: Modeling orbits of celestial bodies.
Conclusion
This article has provided a comprehensive guide to analyzing the circle equation x² + y² + 4x - 6y - 36 = 0. By converting the equation to standard form, we successfully identified the circle's center and radius. Furthermore, we evaluated the given statements, highlighting the importance of completing the square in this process. A solid understanding of circle equations forms a cornerstone for tackling various geometric problems and exploring their diverse applications in real-world scenarios.
By mastering these concepts, you'll unlock a deeper appreciation for the elegance and power of analytical geometry. Remember, the journey of mathematical understanding is a continuous one, and each step you take brings you closer to a more profound appreciation of the world around us.
