Graphing & Analysis: Circle Equation And Point Relationship

Hey guys! Let's dive into some math problems. We're going to tackle two parts here: first, understanding and graphing a circle equation, and second, working with a point and a radius to determine its position. This is all about graphing and analyzing equations and inequalities, so get ready to flex those math muscles!

Part A: Graphing a Circle from its Equation

Alright, let's start with the equation: x² + y² - 12x + 18y - 2 = 0. Our goal here is to get this equation into a form that's easy to visualize, namely, the standard form of a circle's equation. Remember that? It looks like this: (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. So, our first step is completing the square.

To complete the square, we need to group the x terms together, the y terms together, and move the constant term to the other side of the equation. So, let's rearrange our equation like this: (x² - 12x) + (y² + 18y) = 2. Now, we'll complete the square for both the x and y terms. To complete the square for x² - 12x, we take half of the coefficient of the x term (-12), square it ((-12/2)² = 36), and add it to both sides of the equation. Do the same thing for the y terms. Half of 18 is 9, and 9 squared is 81. Add 81 to both sides.

This gives us: (x² - 12x + 36) + (y² + 18y + 81) = 2 + 36 + 81. Now, we can rewrite the expressions in parentheses as perfect squares: (x - 6)² + (y + 9)² = 119. Voila! We've got our equation in standard form. Now, we can easily identify the center of the circle as (6, -9) and the radius as √119. Since the square root of 119 is approximately 10.9, the radius of our circle is about 10.9 units. To graph this, you'd plot the center at (6, -9) and then draw a circle with a radius of approximately 10.9 units.

Remember the steps: rearrange, complete the square, and identify the center and radius. It's a fundamental skill when dealing with circle equations! This process allows us to not only graph the circle but also understand its properties, like its center and radius, which are crucial for further analysis. The ability to transform an equation from general form to standard form is key. This is a common task in many mathematical problems.

Let's get even more specific. The center of the circle is the point (6, -9). This is the exact center around which we'll draw the circle. The radius, which we calculated as the square root of 119, tells us how far from the center every point on the circle is. So, to graph it, imagine a compass: place the pointy end at (6, -9) and then draw a circle with a radius of approximately 10.9 units. And there you have it – a visual representation of your equation! The ability to convert from general to standard form is essential for this. So, practice, practice, practice!

Finding the Center and Radius

The most important takeaway here is how to find the center and radius. This process applies to any circle equation in this general form. The key is completing the square, a technique that transforms the equation into a more manageable form.

So, recap: Identify the center, (h, k), by taking the opposite signs of the numbers within the parentheses in the standard form. Then, find the radius, r, by taking the square root of the constant on the right side of the equation. Remember these two things and you're golden! Being able to visualize and understand these properties is a significant step in your understanding of geometric principles.

Part B: Analyzing a Point's Relationship to the Circle

Now, let's move on to the second part of the problem. We're given a point, (-2, -9), and a radius, R = 2√30. We want to determine the relationship between this point and the circle. To do this, we need to compare the distance from the point to the center of the circle with the radius of the circle.

First, we need to find the distance between the point (-2, -9) and the center of the circle, which we know is (6, -9). We can use the distance formula for this:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Plugging in our values, we get:

Distance = √[(-2 - 6)² + (-9 - (-9))²] = √[(-8)² + (0)²] = √64 = 8

So, the distance between the point (-2, -9) and the center of the circle is 8 units. Now, we compare this distance to the radius, R = 2√30. Let's approximate 2√30. Since √30 is a bit more than 5 (because 5² = 25), 2√30 is a bit more than 10.9, which, as a reminder, is the approximate radius we found in the first part. Let's get the exact value of 2√30 to do a perfect comparison. It's approximately 10.95. Since 8 < 10.95, the distance from the point to the center is less than the radius. This means the point lies inside the circle.

So the conclusion is the point lies within the interior of the circle. Visualizing this can be super helpful – imagine the circle, the center, and then the point. Because the distance from the point to the center is smaller than the radius, the point has to be somewhere inside the circular boundary. It is an inside point.

Determining the Point's Position

The core of this section is about understanding the relationship between the distance from the point to the center of the circle and the radius. By comparing these two values, we can determine whether the point lies inside, outside, or on the circle. This concept is fundamental to many geometry and coordinate geometry problems.

Putting it All Together: The Complete Picture

So, in summary, we've taken an equation and graphed a circle, found the center and radius, and then, with some more given information, analyzed a point's position relative to the circle. We've used completing the square, the distance formula, and our knowledge of circles to solve these problems.

Remember, mastering the concepts of circle equations and point-circle relationships is super important for a good grasp of coordinate geometry. Keep practicing, and you'll become a pro in no time! Keep in mind the significance of each step and how they contribute to solving the problems. The practice of similar exercises will help you a lot to build your skills.

Conclusion: Mastering Circle Geometry

Alright, guys, we've reached the end! We've tackled a circle equation, a point, and some radius magic. You've seen how to take a seemingly complex equation and break it down step-by-step. Remember, practice is key! Review these steps, try similar problems, and don't hesitate to ask questions. Geometry can be a blast, and I hope this helped you feel more confident in your abilities. Keep up the awesome work!