Circle Equation Analysis Unveiling Center, Radius, And Properties

Hey guys! Let's dive into the fascinating world of circles and unravel the mysteries hidden within the equation $x^2+y^2-2x-8=0$. We're going to explore its properties, pinpoint its center, calculate its radius, and ultimately determine which statements about this circle hold true. Buckle up, because we're about to embark on a mathematical adventure!

Decoding the Circle's Equation: Standard Form is Key

To truly understand this circle, our first step involves transforming the given equation into the standard form of a circle's equation. This form, $(x-h)^2 + (y-k)^2 = r^2$, is like a secret code that reveals the circle's center $(h, k)$ and radius $r$ at a single glance. So, how do we crack this code? We'll employ a technique called "completing the square." This method allows us to rewrite the equation in a more insightful format.

Let's start by grouping the $x$ terms together: $(x^2 - 2x) + y^2 - 8 = 0$. Now, focus on the expression within the parentheses, $x^2 - 2x$. To complete the square, we need to add and subtract a specific value. This value is determined by taking half of the coefficient of our $x$ term (-2), squaring it ((-1)^2 = 1), and then adding and subtracting it within the parentheses. So, our equation becomes: $(x^2 - 2x + 1 - 1) + y^2 - 8 = 0$.

Notice how the first three terms within the parentheses, $x^2 - 2x + 1$, now form a perfect square: $(x - 1)^2$. Let's rewrite the equation: $(x - 1)^2 - 1 + y^2 - 8 = 0$. Now, we'll move the constant terms to the right side of the equation: $(x - 1)^2 + y^2 = 9$. Aha! We've successfully transformed the equation into standard form! Now, let's interpret the information it reveals.

By comparing our equation $(x - 1)^2 + y^2 = 9$ with the standard form $(x - h)^2 + (y - k)^2 = r^2$, we can immediately identify the circle's center and radius. The center $(h, k)$ corresponds to $(1, 0)$, and the radius $r$ is the square root of 9, which is 3. So, we've discovered that this circle has a center at (1, 0) and a radius of 3 units. This transformation into standard form is a crucial step in analyzing any circle's equation. It unlocks the essential information needed to understand the circle's properties and behavior. Mastering this technique is fundamental for tackling various circle-related problems in mathematics. Now that we have the standard form, we can confidently analyze the given statements and determine which ones hold true for this particular circle.

Unmasking the Circle's Radius: Is it Really 3 Units?

Now, let's zoom in on the first statement: "The radius of the circle is 3 units." We've already done the legwork to figure this out! Remember when we transformed the equation into the standard form, $(x - 1)^2 + y^2 = 9$? The right side of this equation, 9, represents $r^2$, where $r$ is the radius. To find the radius, we simply take the square root of 9, which indeed gives us 3. So, the statement that the radius of the circle is 3 units is absolutely true!

But why is the radius so important? Well, the radius is a fundamental property of a circle. It defines the circle's size – how far every point on the circle is from its center. Knowing the radius allows us to calculate the circle's circumference (the distance around it) and its area (the space it occupies). The radius also plays a vital role in various geometric constructions and calculations involving circles. Think about drawing a circle with a compass; the distance between the compass's point and the pencil tip is precisely the radius! So, understanding the radius is crucial for grasping a circle's characteristics and its relationship to other geometric figures. We've confirmed the radius for this specific circle, but the concept of the radius applies to all circles, regardless of their size or position in the coordinate plane. The standard form of the equation makes identifying the radius a breeze, highlighting the power of this mathematical representation. Let's move on to investigating the location of the circle's center and see if the remaining statements hold water.

Pinpointing the Center: Does it Reside on the Axes?

Let's shift our focus to the next two statements, which concern the location of the circle's center. The statements are: "The center of the circle lies on the $x$-axis" and "The center of the circle lies on the $y$-axis." Remember, we already determined the center of the circle when we transformed the equation into standard form. We found the center to be at the point (1, 0). Now, let's analyze these statements in light of this information.

To determine if a point lies on the $x$-axis, we need to check its $y$-coordinate. If the $y$-coordinate is 0, then the point lies on the $x$-axis. In our case, the center's coordinates are (1, 0), and the $y$-coordinate is indeed 0. Therefore, the statement "The center of the circle lies on the $x$-axis" is true! The $x$-axis is the horizontal line where all points have a $y$-coordinate of 0. Since our circle's center has a $y$-coordinate of 0, it sits perfectly on this axis. This means the circle is positioned horizontally in the coordinate plane, with its center aligned along the $x$-axis.

Now, let's consider the statement about the $y$-axis. To determine if a point lies on the $y$-axis, we need to check its $x$-coordinate. If the $x$-coordinate is 0, then the point lies on the $y$-axis. However, the center of our circle has an $x$-coordinate of 1, not 0. Therefore, the statement "The center of the circle lies on the $y$-axis" is false. The $y$-axis is the vertical line where all points have an $x$-coordinate of 0. Our circle's center has an $x$-coordinate of 1, placing it one unit to the right of the $y$-axis. So, the circle is not centered vertically along the $y$-axis. This analysis of the center's coordinates demonstrates how we can use the standard form of a circle's equation to pinpoint its exact location in the coordinate plane. By examining the coordinates, we can confidently determine whether the center lies on either the $x$-axis or the $y$-axis.

The Verdict: Which Statements Reign Supreme?

Alright, guys, we've thoroughly dissected the equation $x^2 + y^2 - 2x - 8 = 0$ and uncovered its secrets. We transformed it into standard form, identified its center and radius, and carefully analyzed each statement. Let's recap our findings:

• The radius of the circle is 3 units: TRUE
• The center of the circle lies on the $x$-axis: TRUE
• The center of the circle lies on the $y$-axis: FALSE

Therefore, the three true statements about the circle are: the radius is 3 units, the center lies on the $x$-axis. We successfully navigated the world of circles and equations, guys! By applying the principles of completing the square and understanding the standard form of a circle's equation, we were able to confidently determine the circle's properties and verify the given statements. This journey highlights the power of mathematical tools in unraveling geometric mysteries and gaining a deeper understanding of the world around us. Keep exploring, keep questioning, and keep those mathematical gears turning!

Circle Equation Analysis Center, Radius, and Properties

Consider a circle with the equation $x^2+y^2-2x-8=0$. Which of the following statements are true? (Select three options) Is the radius of the circle 3 units? Does the center of the circle lie on the $x$-axis? Does the center of the circle lie on the $y$-axis? What is the standard form of the equation of this circle?