Equations Of Circles Diameter 12 Units And Center On Y-axis

In the realm of analytic geometry, circles hold a fundamental position. Representing circles using equations allows us to explore their properties and relationships within the Cartesian coordinate system. One essential form for expressing the equation of a circle is the standard form, which provides immediate insights into the circle's center and radius. The standard equation of a circle is given by

(x - h)² + (y - k)² = r²

where (h, k) represents the center of the circle, and r denotes the radius. Understanding this equation is crucial for analyzing and manipulating circles in various mathematical contexts.

This article delves into the specific scenario of circles possessing a diameter of 12 units and centers situated on the y-axis. We will explore how these conditions translate into specific equations using the standard form. By carefully considering the implications of a center lying on the y-axis and a fixed diameter, we can identify the correct equations representing these circles.

Before diving into the problem, it's essential to solidify our understanding of some key concepts:

• Standard Equation of a Circle: The standard equation, as mentioned earlier, is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
• Diameter and Radius: The diameter of a circle is twice its radius. In our case, a diameter of 12 units implies a radius of 6 units.
• Center on the y-axis: If a circle's center lies on the y-axis, its x-coordinate (h) will be 0. This simplifies the standard equation.

The question at hand asks us to identify the equations that represent circles with a diameter of 12 units and a center lying on the y-axis. This seemingly simple statement contains the key to solving the problem. We must carefully dissect each condition and translate it into mathematical constraints on the circle's equation.

Let's break down the given conditions step by step:

1. Diameter of 12 units: This implies that the radius (r) of the circle is 12 / 2 = 6 units. Therefore, r² = 6² = 36.
2. Center on the y-axis: This means the x-coordinate of the center (h) is 0. The center can be represented as (0, k), where k is the y-coordinate.

Now, we can substitute these values into the standard equation of a circle:

(x - 0)² + (y - k)² = 36

Simplifying, we get:

x² + (y - k)² = 36

This is the general form of the equation for a circle that meets our conditions. The only variable left is k, which represents the y-coordinate of the center. The value of k will determine the specific position of the circle's center along the y-axis. The radius, however, is fixed by the diameter and dictates the expansion of the circle from the center point. To ensure clarity, the radius cannot deviate from the calculated value based on the diameter provided. Each variation in the radius affects the final equation. To fully grasp the concept, let us dive deeper into how each element plays a pivotal role in defining the circle.

Now, let's examine the given options and see which ones fit our derived equation form:

• $x^2+(y-3)^2=36$: This equation is in the form x² + (y - k)² = 36, where k = 3. This represents a circle with a center at (0, 3) and a radius of 6. Thus, this option satisfies our conditions. The center lies on the y-axis, and the diameter is indeed 12 units, making it a valid choice.
• $x^2+(y-5)^2=6$: While this equation has the form x² + (y - k)² = some constant, the constant is 6, not 36. This means the radius squared is 6, so the radius is √6, not 6. Therefore, this circle does not have a diameter of 12 units and is not a valid option. The significance of each numerical value and their location within the equation is crucial. To make this circle fulfill the conditions, we need to adjust the numerical constant on the right-hand side of the equation.
• $(x-4)^2+y^3=36$: This is not the equation of a circle because it contains a $y^3$ term. The standard equation of a circle only contains squared terms of x and y. This is a fundamental deviation from the standard equation form, immediately disqualifying this option. Therefore, there's no need to analyze further; it is not a valid equation for a circle under any circumstances.
• $(x+6)^2+y^2=144$: This equation represents a circle with center (-6, 0) and radius 12 (since 144 is 12 squared). While the radius is correct (diameter 12), the center is not on the y-axis because the x-coordinate is -6. Hence, this option is not valid. The center's coordinates significantly dictate whether the circle fulfills the condition of lying on the y-axis. The y-coordinate doesn't matter for this condition, but the x-coordinate must be zero.
• $x^2+(y+8)^2=36$: This equation is in the form x² + (y - k)² = 36, where k = -8. This represents a circle with a center at (0, -8) and a radius of 6. Thus, this option satisfies our conditions. The center lies on the y-axis, and the diameter is indeed 12 units, making it a valid choice. The sign of the constant within the parenthesis matters. The sign change allows us to accurately pinpoint the location of the circle's center in the Cartesian plane.

The two equations that represent circles with a diameter of 12 units and a center that lies on the y-axis are:

• $x^2+(y-3)^2=36$
• $x^2+(y+8)^2=36$

These equations adhere to both the diameter and center conditions outlined in the problem.

We can generalize this approach to solve similar problems involving circles. The key is to:

1. Identify the given conditions (diameter/radius, center location, etc.).
2. Translate these conditions into mathematical constraints on the standard equation of a circle.
3. Substitute the known values into the standard equation.
4. Evaluate the options based on the derived equation and constraints.
5. Ensure every constraint is met to avoid erroneous selections. It is paramount to adhere to the precise requirements specified in the prompt.

For example, if we are asked to find the equation of a circle with a specific radius and a center on the x-axis, we would follow a similar process. The center on the x-axis would imply that the y-coordinate of the center (k) is 0. We would then substitute this information along with the radius into the standard equation and evaluate the given options.

This problem highlights the importance of accurately interpreting mathematical statements. Seemingly small details, such as the diameter being 12 units versus the radius being 12 units, can significantly impact the solution. Similarly, understanding the implications of the center lying on the y-axis is crucial for correctly applying the standard equation of a circle.

Careless interpretation can lead to the selection of incorrect options. For instance, if we misinterpret the diameter as the radius, we would be looking for an equation with a constant term of 144 (12 squared) instead of 36 (6 squared). Such a mistake would lead us down the wrong path. Attention to detail and a thorough understanding of the concepts are essential for achieving accurate results.

By applying our knowledge of the standard equation of a circle and carefully analyzing the given conditions, we were able to identify the equations representing circles with a diameter of 12 units and a center on the y-axis. This problem serves as a good example of how geometric concepts can be translated into algebraic equations and vice versa. The systematic approach we employed can be applied to a wide range of circle-related problems.

Remember, the standard equation (x - h)² + (y - k)² = r² is your friend when working with circles. Mastering its use and understanding the roles of h, k, and r will empower you to solve various circle-related problems with confidence. This mastery is not merely about memorization, but about deeply understanding the interplay between geometry and algebra. This understanding enhances not only problem-solving skills but also a profound appreciation of mathematical beauty and elegance.