Transforming Circle Equations To Standard Form Finding Center And Radius

In the realm of analytic geometry, circles hold a prominent position. Their elegant symmetry and consistent properties make them fundamental shapes in mathematics and various applications. One common task is to analyze the equation of a circle to determine its key features: the center and the radius. When presented with a circle's equation in a general form, such as $x^2 + y^2 + 4x + 12y + 39 = 0$, it's not immediately obvious what the circle's center and radius are. To extract this information, we employ a technique called completing the square. This method allows us to rewrite the equation in the standard form of a circle, which is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the center and $r$ represents the radius.

To effectively tackle this, our initial focus centers on rearranging the provided equation, $x^2 + y^2 + 4x + 12y + 39 = 0$, to group terms containing $x$ and $y$ together. This strategic regrouping sets the stage for the subsequent step of completing the square for both variables. By isolating the $x$ and $y$ terms, we can treat each variable separately, thereby simplifying the process of completing the square. This approach not only clarifies the manipulations required but also allows for a systematic transformation of the equation into the standard form we seek. Therefore, we begin by reorganizing the terms to facilitate the application of the completing the square method.

To begin, we focus on the x-terms: $x^2 + 4x$. To complete the square, we take half of the coefficient of the x-term (which is 4), square it (which gives us $(4/2)^2 = 2^2 = 4$), and add it to the expression. This process ensures that we create a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. In this case, $x^2 + 4x$ becomes $x^2 + 4x + 4$, which can be factored into $(x + 2)^2$. This is a crucial step in transforming the equation into the standard form of a circle. By adding this constant, we are essentially reshaping the expression to reveal the squared term that defines the circle's equation. The key here is to recognize that by completing the square, we are not changing the fundamental equality of the equation; we are simply rewriting it in a more useful form.

Next, we turn our attention to the y-terms: $y^2 + 12y$. Following the same procedure as with the x-terms, we take half of the coefficient of the y-term (which is 12), square it (which gives us $(12/2)^2 = 6^2 = 36$), and add it to the expression. This transforms the expression into a perfect square trinomial. Thus, $y^2 + 12y$ becomes $y^2 + 12y + 36$, which can be factored into $(y + 6)^2$. Completing the square for the y-terms is just as essential as it was for the x-terms. It allows us to isolate another squared term, which, together with the squared x-term, will form the left side of the standard circle equation. By completing the square for both variables, we are effectively structuring the equation to reveal the circle's center and radius, which are embedded within the squared terms. This step is pivotal in converting the general form of the equation into the standard form.

Now, we incorporate the values we added to complete the squares back into the original equation. We added 4 to complete the square for the x-terms and 36 to complete the square for the y-terms. To maintain the equality of the equation, we must add these same values to the right side as well. Thus, we rewrite the equation as $x^2 + 4x + 4 + y^2 + 12y + 36 + 39 = 4 + 36$. This step is crucial because it ensures that the equation remains balanced. Adding the same values to both sides is a fundamental algebraic principle that allows us to manipulate equations without changing their solutions. By including these constants, we are effectively compensating for the terms we added to complete the squares, thus preserving the original relationship expressed by the equation. This careful adjustment is essential for accurately transforming the equation into its standard form.

With the adjustments made, we can rewrite the equation in its completed square form. The left side of the equation now becomes $(x + 2)^2 + (y + 6)^2$, and the right side simplifies to $4 + 36 - 39 = 1$. This transformation is the culmination of the steps we've taken, and it brings the equation into the standard form of a circle: $(x - h)^2 + (y - k)^2 = r^2$. This standard form is the key to unlocking the circle's properties. It directly reveals the center and the radius, which are the fundamental characteristics that define a circle. By reaching this form, we have successfully converted the original equation into a format that makes these properties readily apparent.

The equation is now in the form $(x + 2)^2 + (y + 6)^2 = 1$. Comparing this to the standard form $(x - h)^2 + (y - k)^2 = r^2$, we can identify the center and the radius of the circle. The center $(h, k)$ is given by the values that are subtracted from $x$ and $y$ inside the parentheses. In this case, we have $(x - (-2))^2 + (y - (-6))^2 = 1^2$, so the center is $(-2, -6)$. The radius $r$ is the square root of the constant on the right side of the equation. Here, $r^2 = 1$, so $r = \sqrt{1} = 1$. This step is the payoff for all the work we've done. The standard form of the equation directly provides the circle's center and radius, making them easy to identify. This information is crucial for understanding the circle's position and size in the coordinate plane.

Thus, we have successfully determined that the equation represents a circle with center $(-2, -6)$ and radius $1$. This complete solution demonstrates the power of completing the square as a technique for analyzing conic sections. By transforming the equation into standard form, we were able to easily extract the essential information about the circle. This process not only provides the center and radius but also confirms that the equation indeed represents a circle, as opposed to a degenerate case or another conic section. The ability to manipulate equations and recognize standard forms is a fundamental skill in analytic geometry and is essential for solving a wide range of mathematical problems.

In summary, the given equation $x^2 + y^2 + 4x + 12y + 39 = 0$ represents a circle. By completing the square, we transformed the equation into the standard form $(x + 2)^2 + (y + 6)^2 = 1$. From this, we identified the center of the circle as $(-2, -6)$ and the radius as $1$. This exercise showcases the utility of completing the square in determining the characteristics of circles and other conic sections. The method allows us to move from a general form equation to a standard form, which readily reveals the key properties of the geometric shape. The ability to perform this transformation is a valuable tool in mathematical analysis and problem-solving.