Analyzing The Circle Equation X^2 + Y^2 - 2x - 8 = 0 Radius And Center Determination

Introduction

This article delves into the analysis of a circle's equation, specifically $x^2 + y^2 - 2x - 8 = 0$. We aim to determine several key properties of the circle, including its radius and the location of its center. By converting the given equation into the standard form of a circle's equation, we can easily identify these characteristics. We will then evaluate the provided statements to ascertain which ones accurately describe the circle. Understanding the geometry behind circle equations allows us to solve a variety of problems related to circles, their positions, and their interactions with other geometric figures. This exploration will not only solidify your understanding of circles but also enhance your problem-solving skills in analytic geometry.

Transforming the Equation to Standard Form

The given equation of the circle is $x^2 + y^2 - 2x - 8 = 0$. To determine the circle's center and radius, we need to rewrite this equation in the standard form, which is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the center of the circle and $r$ is the radius. To accomplish this, we will use the method of completing the square. This technique involves manipulating the equation to create perfect square trinomials for both the $x$ and $y$ terms. By completing the square, we transform the equation into a more recognizable form that reveals the circle's key properties.

First, we group the $x$ terms together: $(x^2 - 2x) + y^2 - 8 = 0$. To complete the square for the $x$ terms, we need to add and subtract the square of half the coefficient of the $x$ term. The coefficient of the $x$ term is -2, half of which is -1, and the square of -1 is 1. Therefore, we add and subtract 1 within the parentheses: $(x^2 - 2x + 1 - 1) + y^2 - 8 = 0$. Now, we can rewrite the first three terms as a perfect square: $((x - 1)^2 - 1) + y^2 - 8 = 0$.

Next, we simplify the equation by combining the constant terms: $(x - 1)^2 - 1 + y^2 - 8 = 0$. This simplifies to $(x - 1)^2 + y^2 - 9 = 0$. Now, we isolate the squared terms on one side and the constant term on the other side: $(x - 1)^2 + y^2 = 9$. This is the standard form of the circle's equation. From this form, we can easily identify the center and the radius of the circle.

Identifying the Center and Radius

Now that we have the equation in the standard form $(x - 1)^2 + y^2 = 9$, we can directly read off the center and the radius. Comparing this equation to the standard form $(x - h)^2 + (y - k)^2 = r^2$, we see that $h = 1$, $k = 0$, and $r^2 = 9$. Therefore, the center of the circle is $(1, 0)$, and the radius is $r = \sqrt{9} = 3$ units. Understanding how to extract this information from the standard form of the equation is crucial for solving many problems related to circles. The center and radius are fundamental properties that define the circle's position and size in the coordinate plane.

This process of converting the equation to standard form highlights the elegance of analytic geometry. By manipulating algebraic expressions, we can uncover geometric properties. The center and radius not only define the circle but also serve as key parameters in various geometric calculations and constructions involving circles. Recognizing the significance of the standard form helps in visualizing the circle and its relationships with other geometric elements.

Evaluating the Statements

Having determined the center and radius of the circle, we can now evaluate the given statements:

A. The radius of the circle is 3 units. B. The center of the circle lies on the $x$-axis. C. The center of the circle lies on the $y$-axis.

Based on our calculations, the radius of the circle is indeed 3 units, so statement A is true. The center of the circle is $(1, 0)$. Since the $y$-coordinate of the center is 0, the center lies on the $x$-axis. Thus, statement B is also true. However, since the $x$-coordinate of the center is 1 (not 0), the center does not lie on the $y$-axis, making statement C false. This analysis demonstrates how we can use the derived properties to verify statements about the circle.

Each statement provides a piece of information about the circle's characteristics. Statement A focuses on the size, while statements B and C focus on the location of the center. By carefully examining the center's coordinates, we can determine its position relative to the coordinate axes. This type of problem-solving reinforces the connection between algebraic representations and geometric interpretations.

Conclusion

In summary, by converting the given equation $x^2 + y^2 - 2x - 8 = 0$ into the standard form $(x - 1)^2 + y^2 = 9$, we identified the center of the circle as $(1, 0)$ and the radius as 3 units. Based on these findings, statements A and B are true, while statement C is false. This exercise demonstrates the importance of understanding the standard form of a circle's equation and how it allows us to easily extract key information about the circle. The ability to manipulate equations and interpret geometric properties is a fundamental skill in mathematics.

This exploration highlights the power of analytic geometry in bridging the gap between algebra and geometry. By applying algebraic techniques, we can analyze and understand geometric figures more effectively. The circle, as a fundamental geometric shape, is often used as a building block in more complex geometric constructions and problems. Therefore, a thorough understanding of circles and their equations is essential for further studies in mathematics and related fields.