Circle Equation: Radius And Center Analysis

Let's dive into the world of circles and explore the properties of a circle given its equation. In this article, we will analyze the circle defined by the equation $x^2 + y^2 - 2x - 8 = 0$. Our goal is to determine key characteristics such as its radius and center, and ultimately, identify which statements about the circle are true. So, grab your thinking caps, and let's embark on this geometrical journey!

Understanding the Standard Equation of a Circle

To effectively analyze the given circle equation, we first need to understand the standard form of a circle's equation. The standard equation of a circle with center $(h, k)$ and radius $r$ is given by:

$(x - h)^2 + (y - k)^2 = r^2$

This equation is derived from the Pythagorean theorem, where the distance between any point $(x, y)$ on the circle and the center $(h, k)$ is equal to the radius $r$. By rewriting the given equation in this standard form, we can easily identify the center and radius of our circle.

Completing the square is a crucial technique here. We need to manipulate the given equation to resemble the standard form. This involves grouping the $x$ terms and $y$ terms, and then adding and subtracting appropriate constants to create perfect square trinomials. This process allows us to rewrite the equation in a form that directly reveals the circle's center and radius. By mastering this technique, you'll be able to tackle a wide range of circle-related problems with confidence.

Transforming the Given Equation

Our given equation is $x^2 + y^2 - 2x - 8 = 0$. To transform this into standard form, we'll complete the square for the $x$ terms. Notice that there is no $y$ term, which means the $y$-coordinate of the center will be 0. Let's rewrite the equation, grouping the $x$ terms:

$(x^2 - 2x) + y^2 = 8$

Now, we need to complete the square for the expression $x^2 - 2x$. To do this, we take half of the coefficient of the $x$ term (-2), square it ((-1)^2 = 1), and add it to both sides of the equation:

$(x^2 - 2x + 1) + y^2 = 8 + 1$

The expression inside the parenthesis is now a perfect square trinomial, which can be factored as $(x - 1)^2$. So, our equation becomes:

$(x - 1)^2 + y^2 = 9$

Now, we can clearly see that the equation is in the standard form $(x - h)^2 + (y - k)^2 = r^2$.

This transformation is the heart of solving this problem. By rewriting the equation in standard form, we unlock the secrets of the circle – its center and radius. Without this step, it would be much harder to determine the correct statements. The ability to manipulate equations and recognize patterns is a fundamental skill in mathematics, and this example beautifully illustrates its importance.

Identifying the Center and Radius

By comparing our transformed equation $(x - 1)^2 + y^2 = 9$ with the standard form $(x - h)^2 + (y - k)^2 = r^2$, we can easily identify the center and radius of the circle.

The center of the circle is $(h, k) = (1, 0)$. The radius squared is $r^2 = 9$, so the radius is $r = \sqrt{9} = 3$ units.

Therefore, the center of the circle is at the point (1, 0) and its radius is 3 units. This information is crucial for evaluating the given statements.

The radius and center are the fundamental properties that define a circle. Knowing these values allows us to visualize the circle's position and size in the coordinate plane. It's like having the blueprint of the circle – we can now answer any question about its location and dimensions.

Evaluating the Statements

Now that we know the center and radius, let's evaluate the given statements:

A. The radius of the circle is 3 units. - True, as we calculated. B. The center of the circle lies on the $x$-axis. - True, since the $y$-coordinate of the center (1, 0) is 0, it lies on the x-axis. C. The center of the circle lies on the $y$-axis. - False, since the $x$-coordinate of the center (1, 0) is 1, it does not lie on the y-axis.

Therefore, the three true statements are A and B.

This step highlights the importance of accurate calculations and careful interpretation. We used the information we derived from the equation to directly assess the validity of each statement. This logical process of deduction is a cornerstone of mathematical problem-solving.

Conclusion

In conclusion, by transforming the given equation $x^2 + y^2 - 2x - 8 = 0$ into standard form, we were able to determine that the circle has a center at (1, 0) and a radius of 3 units. This allowed us to identify the correct statements about the circle. This exercise demonstrates the power of algebraic manipulation and the importance of understanding the relationship between a circle's equation and its geometric properties. Remember, practice makes perfect, so keep exploring and solving more circle problems!

For further learning and exploration of circles and their properties, you can visit Khan Academy's Circle Geometry Section. This resource offers a wealth of information, including videos, practice exercises, and articles, to deepen your understanding of this fascinating topic.