Circle Equation Analysis Radius Center And True Statements For X² + Y² - 2x - 8 = 0

July 11, 2025 by Scholario Team 84 views

Exploring the Circle Equation x² + y² - 2x - 8 = 0 A Comprehensive Analysis

Let's delve into the fascinating world of circles and explore the equation x² + y² - 2x - 8 = 0. This equation represents a circle in the Cartesian coordinate system, and by carefully analyzing it, we can uncover key properties such as its radius, center, and its relationship with the coordinate axes. In this comprehensive analysis, we will dissect the equation, transform it into its standard form, and then extract valuable information about the circle's characteristics. This understanding will allow us to determine the truthfulness of the given statements and gain a deeper appreciation for the geometry of circles.

Unveiling the Circle's Properties: Radius and Center

To truly understand the circle represented by the equation x² + y² - 2x - 8 = 0, we need to transform it into its standard form. The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) represents the coordinates of the center and r represents the radius. This form makes it easy to identify the circle's key characteristics.

Let's embark on the transformation process. We begin by grouping the x terms together: (x² - 2x) + y² - 8 = 0. To complete the square for the x terms, we need to add and subtract (2/2)² = 1 inside the parentheses. This gives us (x² - 2x + 1 - 1) + y² - 8 = 0. Now, we can rewrite the expression as (x² - 2x + 1) - 1 + y² - 8 = 0. The expression inside the parentheses is a perfect square, so we can rewrite it as (x - 1)² - 1 + y² - 8 = 0.

Next, we move the constant terms to the right side of the equation: (x - 1)² + y² = 9. Now, we have the equation in standard form: (x - 1)² + (y - 0)² = 3². Comparing this to the standard form (x - h)² + (y - k)² = r², we can immediately identify the center and radius of the circle.

The center of the circle is (h, k) = (1, 0), and the radius is r = 3 units. This tells us that the circle is centered at the point (1, 0) on the Cartesian plane and has a radius of 3 units. We've successfully extracted the fundamental properties of the circle from its equation.

Analyzing the Statements: A Truth Test

Now that we know the circle's center and radius, we can evaluate the given statements and determine their truthfulness. Let's examine each statement individually:

Statement 1: The radius of the circle is 3 units.

Based on our transformation of the equation into standard form, we found that the radius of the circle is indeed 3 units. Therefore, this statement is TRUE. The radius, being a fundamental property of the circle, dictates its size and extent.

Statement 2: The center of the circle lies on the x-axis.

The center of the circle is (1, 0). Since the y-coordinate of the center is 0, the center lies on the x-axis. Therefore, this statement is TRUE. The x-axis is defined as the line where all points have a y-coordinate of 0, so any point with a y-coordinate of 0 lies on this axis.

Statement 3: The center of the circle lies on the y-axis.

The center of the circle is (1, 0). Since the x-coordinate of the center is 1 (not 0), the center does not lie on the y-axis. The y-axis is defined as the line where all points have an x-coordinate of 0. Therefore, this statement is FALSE. The center's position relative to the y-axis is determined by its x-coordinate.

Conclusion: Unveiling the Circle's Secrets

In conclusion, by transforming the equation x² + y² - 2x - 8 = 0 into its standard form, we were able to determine that the circle has a center at (1, 0) and a radius of 3 units. This analysis allowed us to confidently assess the given statements. The true statements are:

The radius of the circle is 3 units.
The center of the circle lies on the x-axis.

This exercise highlights the power of algebraic manipulation in revealing geometric properties. By understanding the standard form of a circle's equation, we can readily extract information about its center, radius, and position in the coordinate plane. This knowledge is crucial for solving a wide range of geometric problems and developing a deeper understanding of the relationship between algebra and geometry.

What are the key characteristics of a circle defined by the equation x² + y² - 2x - 8 = 0? To answer this, we'll embark on a journey of algebraic manipulation and geometric interpretation. Our goal is to transform the given equation into the standard form of a circle's equation, which will allow us to easily identify the circle's center and radius. This process involves completing the square, a powerful technique for rewriting quadratic expressions. Once we have the standard form, we can then analyze the given statements about the circle and determine which ones are true. This exploration will not only provide us with the specific answers but also deepen our understanding of circles and their equations.

Transforming the Equation: Completing the Square

The journey begins with the equation x² + y² - 2x - 8 = 0. Our mission is to rewrite this equation in the standard form (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r represents its radius. This transformation involves a technique called completing the square.

First, let's group the x terms together: (x² - 2x) + y² - 8 = 0. Now, we focus on the expression inside the parentheses. To complete the square, we need to add and subtract a specific value. This value is determined by taking half of the coefficient of the x term (-2), squaring it ((-2/2)² = 1), and adding and subtracting it within the parentheses. This ensures that we don't change the overall value of the equation.

So, we rewrite the equation as (x² - 2x + 1 - 1) + y² - 8 = 0. Notice that we've added and subtracted 1. The expression x² - 2x + 1 is now a perfect square trinomial, which can be factored as (x - 1)². The equation now becomes (x - 1)² - 1 + y² - 8 = 0.

Next, we simplify the equation by moving the constant terms to the right side: (x - 1)² + y² = 9. This equation is now in the standard form of a circle's equation. We can rewrite 9 as 3², giving us (x - 1)² + (y - 0)² = 3². The 'y²' term can be thought of as (y - 0)², making it clear that the y-coordinate of the center is 0.

By completing the square, we've successfully transformed the original equation into its standard form. This form reveals the circle's center and radius, which are crucial for understanding its properties.

Extracting the Circle's Characteristics: Center and Radius

With the equation in standard form, (x - 1)² + (y - 0)² = 3², extracting the circle's characteristics becomes a straightforward task. By comparing this equation to the general standard form (x - h)² + (y - k)² = r², we can identify the values of h, k, and r.

The center of the circle is given by the coordinates (h, k). In our equation, h = 1 and k = 0. Therefore, the center of the circle is (1, 0). This point represents the circle's central location on the Cartesian plane. The radius of the circle is given by r. In our equation, r² = 3², so r = 3. This means the circle has a radius of 3 units. The radius determines the circle's size, measuring the distance from the center to any point on the circle's circumference.

Having determined the center and radius, we now have a clear picture of the circle's geometry. We know its precise location and its overall size. This knowledge is essential for analyzing the given statements about the circle and determining their truthfulness.

Evaluating the Statements: A Geometric Detective

Now that we have the circle's center (1, 0) and radius 3, we can put on our geometric detective hats and evaluate the given statements. Each statement makes a claim about the circle's properties, and we will use our knowledge to determine whether the claim is true or false.

Statement 1: The radius of the circle is 3 units.

This statement directly addresses the radius of the circle. We have already determined that the radius is indeed 3 units by analyzing the standard form of the equation. Therefore, this statement is TRUE. The radius is a fundamental characteristic of the circle, and our calculations confirm this statement.

Statement 2: The center of the circle lies on the x-axis.

This statement concerns the location of the circle's center relative to the coordinate axes. The center of the circle is (1, 0). Points on the x-axis have a y-coordinate of 0. Since the center's y-coordinate is 0, it lies on the x-axis. Therefore, this statement is TRUE. The x-axis serves as a horizontal reference line, and the circle's center is situated directly on it.

Statement 3: The center of the circle lies on the y-axis.

This statement makes a similar claim about the y-axis. Points on the y-axis have an x-coordinate of 0. The center of the circle is (1, 0), which has an x-coordinate of 1 (not 0). Therefore, the center does NOT lie on the y-axis. This statement is FALSE. The y-axis serves as a vertical reference line, and the circle's center is located to the right of it.

Conclusion: Unlocking the Secrets of the Circle

In conclusion, by transforming the equation x² + y² - 2x - 8 = 0 into its standard form, we have successfully unlocked the secrets of the circle. We determined that the circle has a center at (1, 0) and a radius of 3 units. Armed with this information, we evaluated the given statements and found the following to be true:

The radius of the circle is 3 units.
The center of the circle lies on the x-axis.

This exercise demonstrates the power of algebraic techniques in revealing geometric properties. Completing the square allowed us to rewrite the equation in a form that directly revealed the circle's center and radius. This knowledge, in turn, enabled us to confidently analyze the given statements and gain a deeper understanding of the circle's characteristics and its relationship to the coordinate axes. The interplay between algebra and geometry is a powerful tool for problem-solving and mathematical exploration.

The equation x² + y² - 2x - 8 = 0 presents us with a fascinating challenge: to decipher the properties of the circle it represents and to determine the validity of several statements concerning it. This involves a multi-step process, starting with the transformation of the equation into its standard form. The standard form, (x - h)² + (y - k)² = r², is the key to unlocking the circle's secrets, as it directly reveals the center (h, k) and the radius r. To achieve this transformation, we will employ the technique of completing the square. Once we have the standard form, we can then analyze each statement, comparing its claims to the circle's known characteristics. This exercise is not just about finding the right answers; it's about developing a deeper understanding of circles, their equations, and the powerful connection between algebra and geometry.

The Quest for Standard Form: Completing the Square

The journey to understanding the circle described by x² + y² - 2x - 8 = 0 begins with transforming the equation into its standard form: (x - h)² + (y - k)² = r². This form provides a clear view of the circle's center (h, k) and radius r. The technique we'll use to achieve this transformation is called completing the square.

Our first step is to group the x terms together: (x² - 2x) + y² - 8 = 0. Next, we focus on the expression within the parentheses. To complete the square, we need to add and subtract a specific value that will allow us to rewrite the expression as a perfect square trinomial. This value is calculated by taking half of the coefficient of the x term (-2), squaring it ((-2/2)² = 1), and adding and subtracting it inside the parentheses. This ensures that we maintain the equation's balance.

We now rewrite the equation as (x² - 2x + 1 - 1) + y² - 8 = 0. Notice that we've added and subtracted 1. The expression x² - 2x + 1 is a perfect square trinomial and can be factored as (x - 1)². This gives us (x - 1)² - 1 + y² - 8 = 0.

To simplify further, we move the constant terms to the right side of the equation: (x - 1)² + y² = 9. We can express 9 as 3², resulting in the equation (x - 1)² + (y - 0)² = 3². This is the standard form of the circle's equation. By completing the square, we've successfully unveiled the circle's fundamental characteristics.

Deciphering the Circle's Code: Center and Radius Revealed

Having transformed the equation into the standard form (x - 1)² + (y - 0)² = 3², we can now easily decipher the circle's key properties: its center and radius. The standard form (x - h)² + (y - k)² = r² provides a direct comparison, allowing us to identify the values of h, k, and r.

The center of the circle is represented by the coordinates (h, k). In our equation, h = 1 and k = 0. Therefore, the circle's center is located at the point (1, 0). This is the central point around which the circle is drawn. The radius of the circle is represented by r. In our equation, r² = 3², which means r = 3. The circle's radius is 3 units, determining the distance from the center to any point on the circle's circumference.

With the center and radius determined, we have a complete picture of the circle's geometry. We know its precise location in the Cartesian plane and its overall size. This knowledge is crucial for evaluating the statements about the circle and determining their truthfulness.

Testing the Truth: Analyzing the Statements

With the circle's center (1, 0) and radius 3 firmly in our grasp, we can now embark on the final stage: analyzing the given statements and determining their validity. Each statement makes a claim about the circle, and we will use our knowledge to assess its truthfulness.

Statement 1: The radius of the circle is 3 units.

This statement directly addresses the circle's radius. We have already calculated the radius to be 3 units by analyzing the standard form of the equation. Therefore, this statement is TRUE. The radius is a fundamental property of the circle, and our calculations confirm its value.

Statement 2: The center of the circle lies on the x-axis.

This statement pertains to the location of the circle's center relative to the coordinate axes. The center of the circle is (1, 0). Points on the x-axis have a y-coordinate of 0. Since the center's y-coordinate is 0, it lies on the x-axis. Therefore, this statement is TRUE. The x-axis serves as a horizontal reference line, and the circle's center is situated directly on it.

Statement 3: The center of the circle lies on the y-axis.

The Verdict: Unveiling the True Statements

In conclusion, by transforming the equation x² + y² - 2x - 8 = 0 into its standard form, we have successfully determined the circle's center and radius. This allowed us to analyze the given statements and identify the true ones. The true statements are:

The radius of the circle is 3 units.
The center of the circle lies on the x-axis.

This exercise highlights the power of algebraic manipulation in unraveling geometric properties. Completing the square allowed us to rewrite the equation in a form that directly revealed the circle's center and radius. This knowledge, in turn, enabled us to confidently assess the given statements and gain a deeper understanding of the circle's characteristics and its relationship to the coordinate axes. The interplay between algebra and geometry is a powerful tool for problem-solving and mathematical exploration.