Circle Equation: Center (5,5), Radius 3cm - Explained!

Hey guys! Let's dive into this math problem and figure out the equation of a circle. We're given some specific details: the center of the circle is at point O(5,5), the radius is 3 cm, and there's this tricky plane intersecting the x-axis. Don't worry, we'll break it down step by step so it's super clear. We will mainly focus on understanding how to formulate the equation of a circle given its center and radius, and then briefly discuss the extra information provided in the problem.

Understanding the Basic Circle Equation

First things first, let's nail the basics. The general equation of a circle in a 2D plane is your best friend here. It looks like this:

(x - h)² + (y - k)² = r²

Where:

• (h, k) represents the coordinates of the center of the circle.
• r is the radius of the circle.

This equation basically comes from the Pythagorean theorem! Think about it: any point (x, y) on the circle is a distance of 'r' away from the center (h, k). That distance formula is just a rearranged version of the Pythagorean theorem. It’s crucial to understand this foundation because it is applied in various contexts, from simple geometry problems to more complex engineering applications. The elegance of this equation lies in its ability to succinctly describe a circle using only the coordinates of its center and its radius, making it a cornerstone in analytic geometry.

The beauty of this equation is its simplicity. It tells us that for any point (x, y) on the circumference of the circle, the square of the horizontal distance from the center (x - h) plus the square of the vertical distance from the center (y - k) will always equal the square of the radius (r²). This relationship is constant for every single point on the circle, and that's what defines it. Understanding this foundational equation opens doors to solving a myriad of problems related to circles, including finding tangents, intersections, and areas.

Mastering this equation is not just about memorizing a formula; it’s about understanding the underlying geometric principles that make it work. It's about seeing the connection between algebra and geometry, and recognizing how a simple equation can describe a fundamental shape. Once you grasp this, you'll find that manipulating and applying this equation in different scenarios becomes much more intuitive and less about rote learning. Think of it as the DNA of the circle – it contains all the essential information needed to define its shape and position in the plane.

Plugging in the Values: Center and Radius

Now, let’s get practical! We know the center of our circle is O(5, 5), which means h = 5 and k = 5. We also know the radius, r, is 3 cm. Let's plug these values into our general equation:

(x - 5)² + (y - 5)² = 3²

Simplifying this, we get:

(x - 5)² + (y - 5)² = 9

And there you have it! This is the equation of our circle. See? Not so scary after all!

This step is where the general formula truly comes to life. By substituting the specific coordinates of the center (5, 5) and the radius (3 cm) into the placeholders (h, k, and r), we transform the abstract equation into a concrete description of our particular circle. Each term in the equation now has a tangible meaning: (x - 5) represents the horizontal distance from any point on the circle to the center, (y - 5) represents the vertical distance, and 9 is the square of the radius, defining the circle's size. This process of substitution is a fundamental skill in mathematics, allowing us to apply general principles to specific situations. It's like having a recipe (the general equation) and using it to bake a specific cake (our circle) by adding the right ingredients in the right amounts.

By doing this, we've effectively translated the geometric properties of the circle (its center and radius) into an algebraic expression. This is the power of analytic geometry – it allows us to describe shapes and their relationships using equations, making it possible to analyze and manipulate them using algebraic tools. The equation we've derived not only defines the circle but also provides a means to answer further questions about it, such as whether a particular point lies on the circle or where the circle intersects with other geometric objects.

What About the Extra Information? The Tricky Plane!

Okay, here's where things get a little interesting. The problem throws in some extra information about a vertical plane alpha that intersects the x-axis at 2 cm and forms a 60º angle with the horizontal. This is important, but guess what? For the equation of the circle itself, it's actually extra fluff! This information is more relevant if we were asked to find the intersection of the circle with this plane, or the distance from the center of the circle to the plane. But since we just need the equation of the circle, we can breathe a sigh of relief and ignore it... for now.

Why can we ignore it? Because the equation of a circle in a 2D plane is solely determined by its center and radius. The orientation or position of any other object in space, like this plane, doesn't change the fundamental equation that defines the circle itself. The plane would come into play if we were trying to solve a different problem, one that involves the relationship between the circle and the plane, such as finding the points of intersection or the distance between them. In those cases, we would need to consider the plane's equation and how it interacts with the circle's equation.

Think of it like this: the equation of a pizza doesn't change if you put a slice of cake next to it. The pizza is still the same pizza, defined by its ingredients and size. Similarly, the circle's equation remains the same, regardless of the presence of the plane. This illustrates an important problem-solving skill: identifying relevant information and filtering out what's not needed for the specific question at hand. It’s easy to get bogged down in details, but focusing on the core requirements helps to simplify the process and arrive at the correct solution more efficiently.

Final Answer and Key Takeaways

So, the equation of the circle with center O(5, 5) and radius 3 cm is:

(x - 5)² + (y - 5)² = 9

Boom! We nailed it!

The key takeaways here are:

1. Know the general equation of a circle: (x - h)² + (y - k)² = r²
2. Identify the center (h, k) and radius (r) from the problem.
3. Plug those values into the equation and simplify.
4. Don't let extra information distract you if it's not needed for the specific question.

Remember, math problems often try to throw you off with extra details. The trick is to identify what's essential to solve the problem and focus on that. In this case, the plane information was a red herring – it wasn't necessary for finding the circle's equation.

This problem highlights the importance of understanding the fundamental concepts before diving into complex scenarios. By starting with the basic equation of a circle, we were able to easily solve the problem despite the extra information. This approach – building from the basics – is a powerful strategy in mathematics and in problem-solving in general. It allows you to break down complex problems into smaller, more manageable parts, and tackle them one step at a time.

Moreover, this exercise underscores the link between geometry and algebra. The equation of a circle is a perfect example of how algebraic expressions can represent geometric shapes. This connection is fundamental to analytic geometry, which provides the tools to describe and analyze geometric figures using algebraic methods. Understanding this connection not only helps in solving specific problems but also broadens your mathematical perspective and enhances your ability to think critically and creatively.

So, next time you encounter a circle problem, remember the basics, identify the key information, and don't be afraid of a little extra fluff! You've got this!