Finding Points On A Circle A Step-by-Step Guide

Hey everyone! Let's tackle a fun math problem today that involves circles. We're going to figure out which point lies on the circle defined by the equation $(x-3)^2 + (y+4)^2 = 6^2$. Don't worry, it's not as scary as it looks! We'll break it down step by step so you can see exactly how to solve this type of problem. Circles might seem complex, but they follow a pretty straightforward formula. Understanding this formula is key to solving these problems. Let's dive in and make sure you're a circle-solving pro!

Understanding the Circle Equation

Before we jump into the options, let's quickly recap the standard equation of a circle. The general form is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius. This equation is your best friend when dealing with circle-related problems, so make sure you're super familiar with it!

In our case, the equation is $(x-3)^2 + (y+4)^2 = 6^2$. So, what does this tell us? Well, we can see that:

• The center of the circle, $(h, k)$, is $(3, -4)$. Notice that the signs are flipped from the equation!
• The radius, $r$, is $6$ because $6^2$ is on the right side of the equation. Knowing the center and radius gives us a complete picture of our circle. We know exactly where it's located on the coordinate plane and how big it is. This information is crucial for determining which points lie on the circle. The center serves as the anchor, and the radius dictates how far away from the center any point on the circle can be. So, with these two pieces of information, we're well-equipped to tackle the problem at hand. The equation is really just a way of expressing the Pythagorean theorem in terms of the circle's center and radius. It states that for any point on the circle, the distance squared from that point to the center is equal to the radius squared. This is a fundamental concept in understanding circles and their equations. So keep this in mind as we move forward to solving the problem at hand.

Testing the Points: The Key to Finding the Right Answer

Now comes the fun part – testing each point to see if it fits our circle equation! The main idea here is that if a point lies on the circle, its coordinates must satisfy the equation $(x-3)^2 + (y+4)^2 = 6^2$. In other words, when we plug the $x$ and $y$ values of the point into the equation, the left side should equal the right side (which is $36$). This method is super straightforward: plug in and check! It's a bit like having a secret code for the circle, and we're seeing which of the given points knows the code. If the code (equation) works for the point, then that point is part of the circle's club. If not, we move on to the next point. This process might seem a little tedious, but it’s a very reliable way to solve this kind of problem. It's also a great way to reinforce your understanding of how the equation represents the circle's properties. So let’s get started and see which point makes the cut!

We'll go through each option one by one:

A. $(9, -2)$

Let's plug $x = 9$ and $y = -2$ into the equation:

$(9 - 3)^2 + (-2 + 4)^2 = 6^2 + 2^2 = 36 + 4 = 40$

Since $40 e 36$, point A does not lie on the circle. It's outside the circle's boundary. When we calculated the result, we got a value greater than 36, which means the distance from this point to the center of the circle is more than the radius. Imagine drawing a line from the center of the circle to this point; it would be longer than the radius, placing the point outside the circle. So, we can confidently say that (9, -2) is not the correct answer.

B. $(0, 11)$

Now let's try $x = 0$ and $y = 11$:

$(0 - 3)^2 + (11 + 4)^2 = (-3)^2 + (15)^2 = 9 + 225 = 234$

Clearly, $234 e 36$, so point B is also not on the circle. This point is way outside the circle! The value we got, 234, is significantly larger than 36, indicating that the distance from this point to the center is much greater than the radius. If you were to plot this point and the circle on a graph, you'd see that the point is quite far away from the circle's edge. Therefore, we can rule out (0, 11) as the correct answer.

C. $(3, 10)$

Let's plug in $x = 3$ and $y = 10$:

$(3 - 3)^2 + (10 + 4)^2 = 0^2 + 14^2 = 0 + 196 = 196$

Again, $196 e 36$, so point C is not on the circle. This point is also quite a distance away from the circle. The result of 196 is much larger than 36, which means the point (3, 10) is located far from the circle's circumference. If we think about it in terms of distance, the length of the line connecting this point to the center of the circle is much greater than the circle’s radius. So, we can eliminate (3, 10) from our list of possible solutions.

D. $(-9, 4)$

Time for $x = -9$ and $y = 4$:

$(-9 - 3)^2 + (4 + 4)^2 = (-12)^2 + 8^2 = 144 + 64 = 208$

Nope, $208 e 36$, so point D is not on the circle either. We're getting closer, but not quite there yet! The calculated value of 208 is far from our target of 36, suggesting that the point (-9, 4) is located at a considerable distance from the circle's edge. If you were to visualize this on a coordinate plane, you would see a significant gap between the point and the circle. So, we can confidently say that (-9, 4) is not the point we're looking for.

E. $(-3, -4)$

Finally, let's try $x = -3$ and $y = -4$:

$(-3 - 3)^2 + (-4 + 4)^2 = (-6)^2 + 0^2 = 36 + 0 = 36$

Bingo! $36 = 36$, so point E does lie on the circle. We found our winner!

The Correct Answer: E. $(-3, -4)$

After meticulously testing each point, we've found that the point (-3, -4) lies on the circle represented by the equation $(x-3)^2 + (y+4)^2 = 6^2$. This process demonstrates a fundamental approach to solving problems involving circles: understanding the equation and applying it to test potential solutions. By plugging in the coordinates of each point into the equation, we were able to determine which point satisfied the equation and therefore lies on the circle. This method is not only effective but also provides a clear and logical way to arrive at the correct answer. Remember, practice makes perfect, so the more you work with circle equations and points, the more comfortable you'll become with these types of problems. So keep practicing, and you'll be a circle-solving expert in no time!

Final Thoughts and Tips

So, there you have it! We successfully navigated the world of circles and found our point. Remember, the key to these problems is:

1. Understanding the circle equation.
2. Knowing how to substitute values into the equation.
3. Systematically testing each option.

These problems can seem intimidating at first, but breaking them down into smaller steps makes them much more manageable. Don’t be afraid to get your hands dirty with the calculations – that’s how you really learn! And remember, practice makes perfect. The more you work with these equations and points, the more comfortable you'll become with them. Keep up the great work, and you'll be mastering circles in no time!

If you encounter similar problems in the future, just remember these steps, and you'll be well-equipped to tackle them. And don't forget, math can be fun! It's like solving a puzzle, and the satisfaction you get from finding the right answer is totally worth it. So keep exploring, keep learning, and most importantly, keep enjoying the journey of mathematical discovery.