Circle Equation Analysis: Radius, Center, And Tangency
Alright guys, let's dive into the fascinating world of circles and their equations! We're going to break down how to analyze a circle's properties, specifically focusing on its radius, center, and how it interacts with other circles – like whether they're tangent or not. We will be analyzing the circle C1 represented by the equation x² + y² – 6y = 0. Let's unravel this step by step.
Unveiling the Circle C1: Radius and Center
First, let's talk about the basics: the radius and center of a circle. These are the fundamental properties that define a circle's size and position on a coordinate plane. To find these, we'll use a little trick called "completing the square." This method allows us to rewrite the given equation in the standard form of a circle's equation, which makes identifying the radius and center a breeze.
The equation we're working with is x² + y² – 6y = 0. To complete the square, we need to focus on the terms involving 'y'. We have y² – 6y. Think of this as part of a perfect square trinomial. Remember, a perfect square trinomial looks like (y – a)² = y² – 2ay + a². So, we need to figure out what 'a' is in our case. We see that 2a corresponds to 6, which means a = 3. Therefore, a² = 9. We add 9 to complete the square, but to maintain the balance of the equation, we also add it to the other side. It’s like adding the same weight to both sides of a scale – keeps everything even!
Our equation now becomes: x² + y² – 6y + 9 = 9. We can rewrite the 'y' terms as a squared term: x² + (y – 3)² = 9. Now, this looks much more familiar! This is the standard form of a circle's equation: (x – h)² + (y – k)² = r², where (h, k) is the center of the circle and 'r' is the radius. Comparing our equation to the standard form, we can immediately see that the center of circle C1 is (0, 3) because h = 0 and k = 3. The radius squared, r², is 9, so the radius 'r' is the square root of 9, which is 3. Bingo! We've found the center and radius of C1.
To summarize, by completing the square, we transformed the original equation into the standard form. From this, we easily identified that the center of C1 is indeed the point (0, 3) and the radius is 3 units. So, the first two statements in the problem are correct. Understanding this process is crucial, guys, because it allows us to decode the information hidden within the equation and visualize the circle in the Cartesian plane. This foundation will be key as we move on to analyzing the tangency with another circle.
Tangency Test: Circle C1 and the Second Circle
Now, let's crank up the challenge a notch! We need to determine if circle C1 is tangent to another circle, which we'll call C2. Circle C2 is defined by the equation x² + y² – 10x – 6y + 30 = 0. To figure out tangency, we'll first find the center and radius of C2, and then analyze the distance between the centers of the two circles in relation to their radii. Remember, two circles are tangent if the distance between their centers is equal to the sum or the absolute difference of their radii. This is a crucial concept in circle geometry, guys!
Just like with C1, we'll use the method of completing the square to rewrite the equation of C2 in standard form. This will allow us to easily identify its center and radius. The equation for C2 is x² + y² – 10x – 6y + 30 = 0. Let's tackle the 'x' and 'y' terms separately. For the 'x' terms (x² – 10x), we need to add (10/2)² = 25 to complete the square. For the 'y' terms (y² – 6y), we already know from our analysis of C1 that we need to add (6/2)² = 9 to complete the square. So, we'll add 25 and 9 to both sides of the equation.
Our equation transforms into: x² – 10x + 25 + y² – 6y + 9 = -30 + 25 + 9. Rewriting this, we get (x – 5)² + (y – 3)² = 4. Aha! Now we can clearly see the center and radius of C2. The center is (5, 3) and the radius is the square root of 4, which is 2. So, we now have the center and radius for both circles: C1 has center (0, 3) and radius 3, while C2 has center (5, 3) and radius 2.
Next, we need to find the distance between the centers of the two circles. We'll use the distance formula: d = √[(x₂ – x₁)² + (y₂ – y₁)²]. Plugging in the coordinates of the centers (0, 3) and (5, 3), we get d = √[(5 – 0)² + (3 – 3)²] = √(5² + 0²) = √25 = 5. So, the distance between the centers is 5 units. Now, we compare this distance to the sum and difference of the radii.
The sum of the radii is 3 + 2 = 5, and the absolute difference is |3 – 2| = 1. We see that the distance between the centers (5) is equal to the sum of the radii (5). This is the key condition for tangency, specifically external tangency. This means the two circles touch each other at exactly one point, and they lie outside of each other. Therefore, the circle defined by x² + y² – 10x – 6y + 30 = 0 is indeed tangent to C1. Great job, team! We've successfully analyzed the relationship between these two circles.
Wrapping Up: Key Insights and Applications
Alright, let's recap what we've learned and think about why this stuff matters. We started with a circle equation and, by using the powerful technique of completing the square, we were able to extract the circle's center and radius. This is a fundamental skill in analytic geometry, guys, and it's super useful in various applications, from computer graphics to physics simulations. The standard form of a circle's equation is your best friend here – memorize it!
Then, we tackled a more complex problem: determining if two circles are tangent. We found that this involves comparing the distance between the centers of the circles to the sum and difference of their radii. Remember, if the distance equals the sum, the circles are externally tangent; if the distance equals the absolute difference, they are internally tangent. If the distance is greater than the sum, they don't intersect; and if it's less than the difference, one circle is contained within the other. This understanding allows us to visualize and analyze geometric relationships in a coordinate plane.
This stuff isn't just abstract math, guys. Think about it: circle equations and tangency concepts are used in GPS systems to calculate distances, in engineering to design gears and wheels, and even in art and design to create aesthetically pleasing patterns and structures. Understanding the relationships between geometric shapes is a core skill in many STEM fields.
So, keep practicing completing the square and analyzing circle equations. It might seem a bit challenging at first, but with consistent effort, you'll become a circle-equation-solving pro! And remember, math is like a puzzle – each piece fits together to reveal a bigger, more beautiful picture. Keep exploring, keep learning, and keep having fun with it!