Finding The Center Of A Circle Solving X²+y²-12x-2y+12=0
Introduction
In the realm of analytic geometry, circles stand as fundamental shapes, each defined by its center and radius. When presented with the equation of a circle, a crucial task involves identifying its center. This article delves into a step-by-step approach to determine the center of a circle given its equation in the general form. Specifically, we will tackle the equation x²+y²-12x-2y+12=0, providing a comprehensive solution and shedding light on the underlying principles. Understanding how to extract key information from a circle's equation is paramount in various mathematical and real-world applications, ranging from graphical representations to solving geometric problems. This exploration will equip you with the necessary skills to confidently tackle similar problems and deepen your understanding of circles in coordinate geometry.
Understanding the General Equation of a Circle
The general equation of a circle serves as the foundation for our analysis. It is expressed as:
x²+y²+2gx+2fy+c=0
Where:
- (-g, -f) represents the center of the circle.
- √(g²+f²-c) denotes the radius of the circle.
This form provides a standardized way to represent any circle in the Cartesian plane. The coefficients of the x and y terms, along with the constant term, hold the key to unlocking the circle's center and radius. By manipulating a given equation into this general form, we can easily extract the coordinates of the center and the length of the radius. This understanding is crucial for solving problems related to circles, including finding distances, determining intersections, and graphing the circle accurately. Furthermore, recognizing the general equation allows us to connect algebraic representations with geometric properties, reinforcing the fundamental link between algebra and geometry.
Transforming the Given Equation to the Standard Form
The heart of solving our problem lies in transforming the given equation, x²+y²-12x-2y+12=0, into the standard form of a circle's equation. This transformation involves a technique called completing the square, a powerful algebraic method used to rewrite quadratic expressions. The goal is to group the x terms and y terms separately and then manipulate them into perfect square trinomials. This process allows us to rewrite the equation in a form that directly reveals the center and radius of the circle. Let's break down the steps:
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Group the x terms and y terms together: (x²-12x) + (y²-2y) + 12 = 0
Here, we rearrange the terms to bring the x terms and y terms into adjacent groups, setting the stage for completing the square.
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Complete the square for the x terms: To complete the square for x²-12x, we take half of the coefficient of the x term (-12), square it ((-6)² = 36), and add it to the expression. To maintain the equation's balance, we also subtract it: (x²-12x+36) - 36
This step transforms the x terms into a perfect square trinomial, which can be factored into (x-6)².
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Complete the square for the y terms: Similarly, for y²-2y, we take half of the coefficient of the y term (-2), square it ((-1)² = 1), and add it to the expression. Again, we subtract it to maintain balance: (y²-2y+1) - 1
This transforms the y terms into a perfect square trinomial, which can be factored into (y-1)².
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Rewrite the equation: Substituting the completed squares back into the equation, we get: (x²-12x+36) + (y²-2y+1) + 12 - 36 - 1 = 0
Now, we have the equation with perfect square trinomials and adjusted constant terms.
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Factor the perfect square trinomials and simplify: (x-6)² + (y-1)² - 25 = 0
This step factors the trinomials into squared binomials, bringing us closer to the standard form.
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Move the constant term to the right side: (x-6)² + (y-1)² = 25
Finally, we isolate the squared terms on one side and the constant term on the other, achieving the standard form of the circle's equation.
This meticulous process of completing the square allows us to transform the general equation into a form that directly reveals the circle's center and radius, making it a cornerstone technique in analytic geometry.
Identifying the Center from the Standard Form
Now that we have successfully transformed the equation into the standard form:
(x-6)² + (y-1)² = 25
We can readily identify the center of the circle. The standard form of a circle's equation is:
(x-h)² + (y-k)² = r²
Where:
- (h, k) represents the center of the circle.
- r represents the radius of the circle.
By comparing our transformed equation with the standard form, we can directly extract the coordinates of the center. In our case:
- h = 6
- k = 1
Therefore, the center of the circle is (6, 1). This straightforward comparison highlights the power of transforming equations into standard forms, as they provide immediate access to key geometric properties.
Conclusion
In conclusion, by employing the technique of completing the square, we successfully transformed the given equation x²+y²-12x-2y+12=0 into the standard form of a circle's equation. This transformation allowed us to readily identify the center of the circle as (6, 1). This process underscores the importance of algebraic manipulation in extracting geometric information. The ability to convert equations into standard forms is a crucial skill in analytic geometry, enabling us to solve a wide range of problems involving circles and other geometric shapes. Mastering these techniques not only enhances our problem-solving abilities but also deepens our understanding of the fundamental connection between algebra and geometry.
