Circle And Point (8, √17) Location Analysis
Hey guys! Let's dive into a cool math problem about circles centered at the origin. We've got a circle that's hanging out perfectly centered at (0,0), and it's got a point at (0, -9) on its edge. The big question we're tackling today is whether the point (8, √17) also chills on this same circle. It's like figuring out if two points belong to the same club, but instead of a velvet rope, we've got the circle's equation to guide us. So, let’s put on our mathematical hats and get started!
Finding the Circle's Radius
So, the first thing we need to do when faced with a circle problem like this is to figure out the radius. Think of the radius as the VIP pass to the circle club – it's the distance from the center of the circle (the heart of the party) to any point on the circle's edge (where all the cool kids are). We know our circle is centered at the origin (0, 0), and it passes through the point (0, -9). This is super helpful because it gives us a direct line to calculate the radius. We can use the distance formula, which is basically the Pythagorean theorem dressed up for coordinate geometry, or we can recognize something even simpler in this case. Since one of our points is at the origin and the other is directly below it on the y-axis, the distance is simply the absolute value of the y-coordinate of the point (0, -9). In other words, we're just counting how many steps down we need to go from the center to reach the point. The radius, in this case, is the absolute value of -9, which is 9. This means our circle is like a perfectly round arena with a 9-unit reach from the center in every direction. Knowing the radius is crucial because it sets the standard – any point that wants to join the circle club needs to be exactly 9 units away from the center. If it's any closer or further, it's not on the circle.
To really nail this down, let's quickly recap why the radius is so important. The radius defines the size of the circle. If you're sketching the circle, the radius tells you how far your compass needs to open. In terms of equations, the radius is a key component. For a circle centered at the origin, the equation looks like x² + y² = r², where r is our radius. This equation is a strict rulebook – any point (x, y) that wants to live on the circle has to satisfy this equation. If a point's coordinates don't fit the equation, it's a no-go. So, now that we know our circle's radius is 9, we're armed with the information we need to determine if any other point belongs to this circle club. We know the bouncer (the equation of the circle) and the secret password (the radius), so let's see if our mystery point, (8, √17), can get in.
Checking if (8, √17) Lies on the Circle
Okay, so we've successfully found our circle's radius, which is a solid 9 units. Now, the million-dollar question is: Does the point (8, √17) belong to this circle? It's like having a potential member for our exclusive club, and we need to check if they meet the criteria. Remember, the golden rule for any point to be on the circle is that its distance from the center must be exactly equal to the radius. If it's too close or too far, it's not in the club. To figure this out, we're going to use the trusty distance formula again. This formula is like our measuring tape for the coordinate plane, telling us precisely how far apart two points are. Given two points (x₁, y₁) and (x₂, y₂), the distance between them is calculated as √[(x₂ - x₁)² + (y₂ - y₁)²]. In our case, one point is the center of the circle (0, 0), and the other point is our potential member, (8, √17). So, we're going to plug these values into the distance formula and see what we get.
Let's break it down step by step. We've got x₁ = 0, y₁ = 0 (the center of the circle), x₂ = 8, and y₂ = √17. Plugging these into the distance formula, we get: Distance = √[(8 - 0)² + (√17 - 0)²]. Now, let's simplify this. (8 - 0)² is simply 8², which is 64. (√17 - 0)² is just (√17)², which conveniently cancels out the square root, giving us 17. So, our equation now looks like: Distance = √(64 + 17). Adding 64 and 17, we get 81. So, now we have: Distance = √81. And the square root of 81? That's a neat 9! So, the distance from the center of the circle to the point (8, √17) is exactly 9 units. But wait, we're not done yet! We need to compare this distance to the radius we calculated earlier. Remember, the radius was also 9 units. Aha! The distance from the center to (8, √17) is the same as the radius. This is like the point showing its VIP pass at the door – it perfectly matches the requirement. Therefore, the point (8, √17) does indeed lie on the circle.
The Equation of the Circle
Now, let's take a step back and look at this problem from a slightly different angle – the equation of the circle. Knowing the equation of a circle is like having a master key that unlocks all the secrets of the circle. For a circle centered at the origin (0, 0), the equation takes a simple and elegant form: x² + y² = r², where 'r' is the radius of the circle. This equation is a powerful tool because it gives us a direct way to check if any point (x, y) lies on the circle. If the coordinates of the point, when plugged into the equation, satisfy the equation (i.e., the left side equals the right side), then the point is on the circle. If not, the point is either inside or outside the circle. We already figured out that our circle has a radius of 9 units. So, if we plug this value into the equation, we get: x² + y² = 9². And since 9² equals 81, the equation of our circle is: x² + y² = 81. This equation is the circle's unique fingerprint – it tells us everything about the circle's size and position in the coordinate plane.
Now, let's use this equation to verify our earlier finding about the point (8, √17). To do this, we're going to substitute the x and y coordinates of the point into the equation and see if it holds true. So, we'll replace x with 8 and y with √17 in the equation x² + y² = 81. This gives us: 8² + (√17)² = 81. Let's simplify this. 8² is 64, and (√17)² is 17 (the square root and the square cancel each other out). So, we have: 64 + 17 = 81. Now, let's add 64 and 17. What do we get? 81! So, our equation becomes: 81 = 81. This is a true statement! The left side of the equation is equal to the right side. This confirms that the point (8, √17) satisfies the equation of the circle. This is like having a double confirmation – we've checked the point's distance from the center and we've checked if it fits the circle's equation, and both methods tell us the same thing: the point (8, √17) is definitely on the circle.
Using the equation of the circle gives us a powerful alternative method to solve this kind of problem. It's like having a backup plan, or a different route to the same destination. In this case, both the distance formula and the circle's equation lead us to the same conclusion, reinforcing our confidence in the answer.
Conclusion
So, let's wrap things up and answer the original question with a resounding sense of mathematical certainty! We started with a circle centered at the origin that contained the point (0, -9). Our mission was to determine if the point (8, √17) also belonged to this circle. We approached this problem with our mathematical toolkit, which included the concept of a circle's radius, the distance formula, and the equation of a circle centered at the origin. First, we figured out the radius of the circle by calculating the distance from the center (0, 0) to the point (0, -9), which gave us a radius of 9 units. This was like setting the stage for our investigation – we knew the size of the circle we were dealing with. Then, we used the distance formula to find the distance from the center (0, 0) to the point (8, √17). We carefully plugged in the coordinates and crunched the numbers, and we found that the distance was also 9 units. This was a promising sign! But we didn't stop there. To be absolutely sure, we also considered the equation of the circle centered at the origin, which is x² + y² = r². We substituted our radius (9) into this equation, giving us x² + y² = 81. This equation became our ultimate test. We plugged in the coordinates of the point (8, √17) into the equation, and to our delight, it satisfied the equation perfectly. This was like getting the final stamp of approval! With both the distance and the equation methods confirming our result, we can confidently say: Yes, the point (8, √17) does indeed lie on the circle.
The correct answer is A: No, the distance from the center to the point (8, √17) is not the same as the radius.
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