Circle Equation Problems: Center (0,0) And Radius/Points

Hey guys! Let's dive into some circle equation problems today. We'll be focusing on circles with their center at the origin (0,0). This makes things a bit simpler, but it's crucial to nail down the basics. We'll tackle finding the equation when we know the radius and when we know a point the circle passes through. So, grab your pencils, and let's get started!

1. Finding the Equation Given the Radius

Okay, so the equation of a circle centered at the origin (0,0) is super straightforward: it's just x² + y² = r², where r is the radius. Remember this, guys, it's the foundation for everything we're doing today!

a. Radius = √5

So, our main keyword here is circle equation, and the first problem gives us a radius of √5. To find the equation, we simply plug this value into our formula: x² + y² = (√5)². What's (√5)²? It's just 5! So, the equation of the circle is x² + y² = 5. See? Easy peasy! Think of it this way, the radius squared represents the distance from the center of the circle to any point on the circle. Squaring the square root of 5 gives us that direct distance value.

Now, let's really break down why this works. The equation x² + y² = r² is actually derived from the Pythagorean theorem. Imagine a right triangle formed by the x-coordinate, the y-coordinate, and the radius as the hypotenuse. The Pythagorean theorem states that a² + b² = c², where a and b are the legs of the triangle, and c is the hypotenuse. In our case, x and y are the legs, and r is the hypotenuse. This geometric interpretation is super important for understanding the equation's origins and for visualizing what's actually going on.

Understanding this connection to the Pythagorean theorem will also help you in more complex problems later on. For example, when the circle isn't centered at the origin, the equation gets a little more complicated, but the underlying principle – the relationship between the coordinates and the distance (radius) – remains the same. So, by grasping the fundamental concept here, you're building a solid foundation for tackling more advanced topics in circle geometry. Remember, guys, math builds upon itself, so mastering the basics is key!

b. Radius = 10

Next up, we have a radius of 10. Using the same circle equation formula, x² + y² = r², we substitute r with 10. This gives us x² + y² = 10², which simplifies to x² + y² = 100. Piece of cake, right? The radius here is a nice, clean whole number, making the squaring operation even simpler. Think about it: a circle with a radius of 10 will be significantly larger than the one with a radius of √5.

Again, let's take a moment to really grasp what this equation represents. x² + y² = 100 means that for any point (x, y) on the circle, the sum of the squares of its coordinates will always equal 100. This constant relationship defines the shape of the circle. Imagine plotting countless points that satisfy this equation – they would all form a perfectly round shape, centered at the origin, with a radius of 10. This visual representation is crucial for truly understanding the concept.

Furthermore, this problem highlights the direct relationship between the radius and the size of the circle. A larger radius directly translates to a larger circle. This might seem obvious, but it's a fundamental understanding that's worth reinforcing. As you encounter more circle problems, you'll start to develop an intuitive sense of how the radius affects the circle's properties, like its circumference and area. So, keep practicing and visualizing these relationships, guys! It'll pay off in the long run.

c. Radius = 8

Okay, last one in this section! We have a radius of 8. Following the circle equation pattern, we get x² + y² = 8². 8 squared is 64, so the equation is x² + y² = 64. You guys are getting the hang of this, I can tell!

Let's compare this circle to the previous ones. It's smaller than the circle with a radius of 10, but larger than the circle with a radius of √5. This reinforces the idea that the radius directly determines the size of the circle. Also, notice how the square of the radius in the equation (64) is not the same as the area of the circle (which would be π * 8² = 64π). It's a common mistake to confuse these two, so it's important to keep them distinct in your mind.

To further solidify your understanding, try thinking about some points that would lie on this circle. For example, the point (8, 0) would definitely be on the circle because 8² + 0² = 64. Similarly, the point (0, 8) would also be on the circle. Can you think of any other points? Try using different values for x and calculating the corresponding y value (or vice versa) that would satisfy the equation. This exercise of finding points on the circle will give you a deeper, more intuitive grasp of the relationship between the equation and the geometric shape it represents.

2. Finding the Equation Given a Point on the Circle

Now, let's switch things up a bit. Instead of the radius, we're given a point that lies on the circle. Remember, the circle equation is still x² + y² = r², but now we need to find r² using the point's coordinates.

a. Point (-3, 5)

We're given the point (-3, 5). This means x = -3 and y = 5. We can plug these values into the equation x² + y² = r² to find r². So, (-3)² + (5)² = r². This simplifies to 9 + 25 = r², which means r² = 34. Therefore, the equation of the circle is x² + y² = 34. See, guys? We're just using the given information and the standard equation to solve for the missing piece!

Let's delve a little deeper into the concept here. The point (-3, 5) lies on the circle, which means its distance from the center (0, 0) is the radius. We're essentially using the distance formula (which is derived from the Pythagorean theorem, remember?) to calculate that distance. By plugging the coordinates into the equation, we're finding the square of the distance, which is exactly what we need for r² in the circle equation.

This method highlights a key property of circles: all points on the circle are equidistant from the center. Knowing a single point on the circle allows us to determine this constant distance (the radius) and thus define the circle's equation. This concept is crucial for understanding various geometric problems involving circles, such as finding tangents, chords, and intersections. So, make sure you're comfortable with this connection between points on the circle, the radius, and the equation. Keep practicing, and it'll become second nature!

b. Point (1, 0)

Next, we have the point (1, 0). Plugging these into our circle equation: 1² + 0² = r². This simplifies to 1 + 0 = r², so r² = 1. Therefore, the equation of the circle is x² + y² = 1. This is a special case – a circle with a radius of 1, often called the unit circle. It's super important in trigonometry and many other areas of math!

The unit circle is a fundamental concept, guys. It's a circle centered at the origin with a radius of 1. Its equation, x² + y² = 1, is incredibly simple, but it has profound implications. For example, the trigonometric functions (sine, cosine, tangent) are often defined in terms of the coordinates of points on the unit circle. Understanding the unit circle is essential for grasping trigonometry and its applications.

Furthermore, the fact that the point (1, 0) lies on this circle should make intuitive sense. It's a point on the x-axis, exactly one unit away from the origin, which is perfectly consistent with the definition of a circle with a radius of 1. This reinforces the connection between the equation, the geometric shape, and the coordinates of points on the circle. So, make sure you have a solid understanding of the unit circle – it's a concept you'll encounter again and again in your mathematical journey.

c. Point (5, -12)

Now we're working with the point (5, -12). Let's plug it into the circle equation: 5² + (-12)² = r². This gives us 25 + 144 = r², so r² = 169. Thus, the equation is x² + y² = 169. This time, we have a larger r² value, indicating a larger circle. Notice how the negative sign on the y-coordinate doesn't affect the calculation because we're squaring it.

This problem provides a good opportunity to emphasize the importance of paying attention to signs. While squaring a negative number results in a positive number, it's crucial to handle negative signs correctly in other mathematical contexts. A small error in a sign can lead to a completely wrong answer. So, always double-check your work and be mindful of the rules of arithmetic.

Also, this example highlights the relationship between the coordinates of a point on the circle and the radius. The point (5, -12) is farther away from the origin than the previous points we've considered. This translates to a larger radius and a larger r² value in the equation. Visualizing these relationships will help you develop a stronger intuition for circle geometry. Try plotting this point and imagining the circle it lies on – it'll make the concept more concrete.

d. Point (√6, 6)

Let's tackle the point (√6, 6). Substituting into x² + y² = r², we get (√6)² + 6² = r². This simplifies to 6 + 36 = r², so r² = 42. The equation of the circle equation is x² + y² = 42. Here, we have a square root involved, but the process remains the same – just square each coordinate and add them up!

This problem introduces a slightly more complex coordinate, a square root. But remember, the process for finding the equation remains identical. This highlights the robustness of the method – it works regardless of the type of numbers involved. This is a crucial point in mathematics: understanding the underlying principles allows you to apply them to a wide range of situations.

Furthermore, dealing with square roots is a common occurrence in geometry and other areas of math. Practice working with them will build your confidence and proficiency. Remember the key properties of square roots, such as (√a)² = a, which we used in this problem. Mastering these basic operations will pave the way for tackling more complex problems involving radicals. So, keep practicing, and don't be intimidated by square roots – they're just another type of number!

e. Point (√2, √3)

Finally, we have the point (√2, √3). Plugging into x² + y² = r², we get (√2)² + (√3)² = r². This simplifies to 2 + 3 = r², so r² = 5. The circle equation is x² + y² = 5. Another example with square roots, but again, we just follow the formula!

This problem is a nice recap of the principles we've been discussing. We're again working with square roots, but the method remains consistent. This reinforces the idea that mathematical concepts are built upon consistent rules and procedures. Once you understand the rules, you can apply them to various situations, even if the numbers look a little different.

Furthermore, this example provides a good opportunity to review the process we've been using throughout this exercise. We're given a point on the circle, we plug its coordinates into the equation x² + y² = r², and we solve for r². This value of r² then becomes part of the final equation of the circle. This step-by-step approach is a valuable problem-solving strategy that can be applied to many mathematical situations. So, remember to break down complex problems into smaller, manageable steps, and you'll be well on your way to success!

Conclusion

So, there you have it, guys! We've tackled finding the equation of a circle centered at the origin, both when given the radius and when given a point on the circle. Remember the key equation: x² + y² = r². Practice these types of problems, and you'll become a circle equation master in no time! Keep up the great work!