Circle Positions: Solving Equations Step-by-Step

Hey guys! Let's dive into some cool math problems. We're going to determine the relative positions of different circles. This involves a bit of algebra, but don't worry, it's totally manageable. We'll be looking at equations and figuring out how these circles relate to each other: do they intersect, touch, or are they completely separate? Let's get started and break it down step-by-step. This is all about understanding circle equations and their geometric interpretations. It's super useful for anyone looking to brush up on their math skills or just curious about how circles work. Understanding these concepts will give you a solid foundation in coordinate geometry, which is super important for higher-level math and even some real-world applications. Ready to unravel the secrets of circle positioning? Let's go!

Circle Equation Fundamentals

Before we jump into the specific problems, let's quickly review the basics. The general equation of a circle is something we need to be very familiar with. It's the key to understanding everything else. The standard form of a circle's equation is: (x - h)^2 + (y - k)^2 = r^2. Here, (h, k) represents the center of the circle, and r is the radius. This simple equation holds all the secrets we need to solve our problems. If we have the equation in this form, we can easily identify the center and radius, which are crucial for determining the circle's position relative to other circles. Now, if the equation is not in the standard form (and often, it isn't!), we need to convert it by completing the square. Completing the square is a technique where you manipulate the equation to get it into the standard form. Remember, the center gives us the location, and the radius gives us the size. Both are important for determining how the circles relate to each other. We are going to make sure that we have this fundamental aspect covered well. So, let’s get this fundamental concept fully understood. Getting a good understanding of this will make the problem so much easier.

Now, let's explore this with examples. If you have (x - 2)^2 + (y + 3)^2 = 25, the center is (2, -3) and the radius is 5. It's that simple! But remember, the equation's standard form is super important. We will look at that later on when we look at our circle equations.

Converting to Standard Form: Completing the Square

So, as we said, what if your equation isn't in standard form? Well, that's where completing the square comes in. It's a handy technique to rewrite any circle equation into the standard form. Let's say we have an equation like this: x^2 + 6x + y^2 - 4y = 12. First, group the x and y terms: (x^2 + 6x) + (y^2 - 4y) = 12. Next, complete the square for both x and y. Take half of the coefficient of x (which is 6), square it (which gives us 9), and add it to both sides. Do the same for y (half of -4 is -2, squared is 4): (x^2 + 6x + 9) + (y^2 - 4y + 4) = 12 + 9 + 4. Now, rewrite the perfect square trinomials: (x + 3)^2 + (y - 2)^2 = 25. Boom! We've got the standard form. The center is (-3, 2), and the radius is 5. Knowing how to convert the form is really important because it makes it easier to tell the relationships between circles.

Determining the Relationship Between Circles

Once we have the equations in standard form, we can determine the relationship between the circles. Here's a quick guide:

• Circles do not intersect: The distance between the centers is greater than the sum of the radii.
• Circles touch externally: The distance between the centers is equal to the sum of the radii.
• Circles touch internally: The distance between the centers is equal to the absolute difference of the radii.
• Circles intersect at two points: The distance between the centers is less than the sum of the radii but greater than the absolute difference of the radii.
• One circle is inside the other: The distance between the centers is less than the absolute difference of the radii. That's a lot, right? But with the help of examples, everything will fall into place.

Analyzing Circle Equations

Okay, let's get down to the actual problems. We are going to analyze each equation and determine the position of the respective circles to each other. This is where it gets fun, and we get to apply our knowledge. Remember our steps: get the standard form if necessary, identify the center and radius, and then, figure out how the circles relate to each other. The whole process becomes clearer when you start applying the methods. Make sure that you understand the concept by practicing it.

Circle 1: $x^2 + y^2 + 8x - 2y - 8 = 0$

First, we need to get this into standard form. Let's complete the square: (x^2 + 8x) + (y^2 - 2y) = 8. For x, we add (8/2)^2 = 16, and for y, we add (-2/2)^2 = 1. So, we have (x^2 + 8x + 16) + (y^2 - 2y + 1) = 8 + 16 + 1. This simplifies to (x + 4)^2 + (y - 1)^2 = 25. The center is (-4, 1), and the radius is 5.

Circle 2: $(x+3)^2 + (y-1)^2 = 16$

This one is already in standard form! The center is (-3, 1), and the radius is 4. Now, we are ready to find the relation between the circles. We have already analyzed the two circles individually. So, let’s go and evaluate their relationship.

Relationship between Circle 1 and Circle 2

To find out, we need the distance between their centers. The distance between (-4, 1) and (-3, 1) is √((-3 - (-4))^2 + (1 - 1)^2) = √(1^2 + 0^2) = 1. The sum of the radii is 5 + 4 = 9. Since the distance (1) is less than the sum of the radii (9), and the distance is also greater than the absolute difference between the radii (|5 - 4| = 1), the circles intersect at two points.

Circle 3: $(x+4)^2 + (y-1)^2 = 4$

This is in standard form. The center is (-4, 1), and the radius is 2.

Relationship between Circle 1 and Circle 3

The distance between (-4, 1) and (-4, 1) is 0. The sum of the radii is 5 + 2 = 7. The absolute difference is |5 - 2| = 3. Since the distance (0) is less than the absolute difference (3), Circle 3 is inside Circle 1.

Relationship between Circle 2 and Circle 3

The distance between (-3, 1) and (-4, 1) is 1. The sum of the radii is 4 + 2 = 6. The absolute difference is |4 - 2| = 2. Since the distance (1) is less than the sum (6) and greater than the absolute difference (2), these circles intersect at two points.

Circle 4: $(x-2)^2 + (y-1)^2 = 1$

Standard form! The center is (2, 1), and the radius is 1.

Relationship between Circle 1 and Circle 4

The distance between (-4, 1) and (2, 1) is 6. The sum of the radii is 5 + 1 = 6. So the circles touch externally.

Relationship between Circle 2 and Circle 4

The distance between (-3, 1) and (2, 1) is 5. The sum of the radii is 4 + 1 = 5. Hence, they touch externally.

Relationship between Circle 3 and Circle 4

The distance between (-4, 1) and (2, 1) is 6. The sum of the radii is 2 + 1 = 3. So the circles do not intersect.

Circle 5: $x^2 + y^2 = 9$

This can be written as (x - 0)^2 + (y - 0)^2 = 9. Center is (0, 0), and the radius is 3.

Relationship between Circle 1 and Circle 5

The distance between (-4, 1) and (0, 0) is √((-4 - 0)^2 + (1 - 0)^2) = √(16 + 1) = √17 ≈ 4.12. The sum of the radii is 5 + 3 = 8. The absolute difference is |5 - 3| = 2. Because the distance is greater than the absolute difference and less than the sum, the circles intersect at two points.

Relationship between Circle 2 and Circle 5

The distance between (-3, 1) and (0, 0) is √((-3 - 0)^2 + (1 - 0)^2) = √(9 + 1) = √10 ≈ 3.16. The sum of the radii is 4 + 3 = 7. The absolute difference is |4 - 3| = 1. Because the distance is greater than the absolute difference and less than the sum, the circles intersect at two points.

Relationship between Circle 3 and Circle 5

The distance between (-4, 1) and (0, 0) is √((-4 - 0)^2 + (1 - 0)^2) = √(16 + 1) = √17 ≈ 4.12. The sum of the radii is 2 + 3 = 5. Because the distance is greater than the sum, the circles do not intersect.

Relationship between Circle 4 and Circle 5

The distance between (2, 1) and (0, 0) is √((2 - 0)^2 + (1 - 0)^2) = √(4 + 1) = √5 ≈ 2.24. The sum of the radii is 1 + 3 = 4. The absolute difference is |1 - 3| = 2. Because the distance is greater than the absolute difference and less than the sum, the circles intersect at two points.

Conclusion: Circle Positioning Summarized

Alright, guys! We've made it through! We've successfully analyzed the relative positions of all the circles. By finding the centers and radii and calculating the distances between the centers, we were able to determine whether the circles intersect, touch, or are completely separate. The most important thing here is practice. Go over these examples and try different circle equations and practice to fully grasp the concepts.

Remember, understanding how to rewrite equations and use distance formulas is key. Coordinate geometry is fun and gives you a good grasp of the geometric concepts. Keep practicing, and you'll become a pro at these circle problems. Good job, everyone!