Circle Equation X²+y²+4x-6y-36=0 Analysis And Conversion
In the realm of analytical geometry, circles hold a fundamental position, and their equations serve as the key to unlocking their geometric properties. This article delves into the intricacies of a specific circle equation, x²+y²+4x-6y-36=0, aiming to provide a comprehensive understanding of its characteristics and how to analyze it effectively. We will explore the process of converting this equation to its standard form, which reveals the circle's center and radius, and then discuss the truthfulness of given statements related to this equation.
Decoding the General Form of a Circle Equation
The given equation, x²+y²+4x-6y-36=0, is presented in the general form of a circle equation, which is expressed as:
x² + y² + Dx + Ey + F = 0
Where D, E, and F are constants. While this form represents a circle, it doesn't immediately reveal the circle's center and radius. To extract this crucial information, we need to transform the equation into its standard form.
Transforming to Standard Form The Power of Completing the Square
The standard form of a circle equation provides a clear representation of the circle's center and radius. It is expressed as:
(x - h)² + (y - k)² = r²
Where (h, k) represents the center of the circle and r represents its radius. To convert the general form equation into this standard form, we employ a technique called "completing the square". This technique allows us to rewrite quadratic expressions as perfect squares, ultimately leading us to the standard form.
Step-by-Step Conversion
Let's embark on the step-by-step process of converting the given equation, x²+y²+4x-6y-36=0, into its standard form:
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Rearrange the terms: Group the x terms and y terms together, and move the constant term to the right side of the equation.
x² + 4x + y² - 6y = 36
This step sets the stage for completing the square for both the x and y terms.
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Complete the square for x terms: To complete the square for the x terms (x² + 4x), we need to add a constant term that will make it a perfect square trinomial. This constant is calculated as (b/2)², where 'b' is the coefficient of the x term. In this case, b = 4, so (b/2)² = (4/2)² = 4. We add this 4 to both sides of the equation to maintain balance.
x² + 4x + 4 + y² - 6y = 36 + 4
The left side now contains a perfect square trinomial for the x terms.
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Complete the square for y terms: Similarly, we complete the square for the y terms (y² - 6y). The coefficient of the y term is -6, so (b/2)² = (-6/2)² = 9. Add 9 to both sides of the equation.
x² + 4x + 4 + y² - 6y + 9 = 36 + 4 + 9
Now, both the x and y terms are expressed as perfect square trinomials.
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Rewrite as squared terms: Factor the perfect square trinomials and simplify the right side of the equation.
(x + 2)² + (y - 3)² = 49
This is the standard form of the circle equation.
Unveiling the Circle's Properties
From the standard form (x + 2)² + (y - 3)² = 49, we can directly identify the circle's center and radius:
- Center: The center of the circle is (h, k) = (-2, 3).
- Radius: The radius of the circle is r = √49 = 7.
Therefore, the circle represented by the equation x²+y²+4x-6y-36=0 has a center at (-2, 3) and a radius of 7.
Assessing the Statements
Now that we've successfully converted the equation to standard form and determined the circle's properties, let's evaluate the given statements:
Statement A To begin converting the equation to standard form, subtract 36 from both sides.
This statement is incorrect. As demonstrated in our step-by-step conversion, the first step involves adding 36 to both sides of the equation to isolate the x and y terms on one side. Subtracting 36 would move the constant term to the left side, hindering the process of completing the square.
Statement B To complete the square for the x terms, add 4 to both sides.
This statement is correct. As we showed in step 2 of the conversion process, adding 4 to both sides is indeed the correct action to complete the square for the x terms. This ensures that the expression x² + 4x can be rewritten as the perfect square (x + 2)².
Conclusion Mastering Circle Equations
By meticulously converting the given circle equation x²+y²+4x-6y-36=0 to its standard form, we successfully determined its center and radius. This process involved the crucial technique of completing the square, a cornerstone of algebraic manipulation. Furthermore, we critically evaluated the provided statements, discerning their truthfulness based on our understanding of the conversion process. This exercise underscores the importance of mastering circle equations and their transformations in analytical geometry. The ability to manipulate these equations empowers us to extract valuable geometric information and solve a wide range of problems involving circles. Remember, the standard form is your key to unlocking the circle's secrets its center and its radius.
To solidify your understanding, try applying these techniques to other circle equations and explore the various properties that can be derived from them. The world of circles is vast and fascinating, and with a solid grasp of the fundamentals, you'll be well-equipped to navigate its intricacies.
