Points On A Circle Centered At The Origin How To Find Them

Determining whether a point lies on a circle is a fundamental concept in coordinate geometry. This article delves into the process of identifying a point on a circle centered at the origin with a radius of 5 units. We'll utilize the distance formula, a crucial tool in this endeavor, and explore the step-by-step methodology to arrive at the correct answer. We will analyze the distance formula: √((x₂ - x₁)² + (y₂ - y₁)²) and apply it to the given points to see which one satisfies the condition of lying on the circle. This exploration will not only reinforce your understanding of circles and distances but also enhance your problem-solving skills in analytical geometry. Understanding this concept is crucial for various applications in mathematics, physics, and engineering, where determining the position of objects relative to a central point is essential. Let's embark on this geometrical journey and unlock the secrets of circles and points!

Understanding the Problem: Circle Centered at the Origin

The core of our problem lies in understanding the properties of a circle centered at the origin. A circle, by definition, is a set of points equidistant from a central point. In our case, this central point is the origin (0, 0) on the Cartesian plane. The radius of the circle, given as 5 units, represents the constant distance between the center and any point on the circle. To determine if a point lies on this circle, we need to verify if its distance from the origin is exactly 5 units. This is where the distance formula comes into play, providing us with a precise method to calculate the distance between two points in a coordinate plane. Visualizing the circle and the points in question can also aid in understanding the problem better. Imagine a circle drawn on a graph with its center at the intersection of the x and y axes, and then picture the given points in relation to this circle. This mental image can often provide an intuitive sense of which points are more likely to be on the circle. The distance formula allows us to move beyond intuition and arrive at a definitive answer.

The Distance Formula: A Key Tool

The distance formula is our primary tool for solving this problem. It provides a way to calculate the distance between any two points in a coordinate plane. The formula is expressed as: √((x₂ - x₁)² + (y₂ - y₁)²), where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. In our specific scenario, one of the points is always the origin (0, 0), and the other point is one of the given options (A, B, C, or D). Therefore, the formula simplifies to √(x² + y²), where (x, y) represents the coordinates of the point we are testing. To determine if a point lies on the circle, we substitute its coordinates into this simplified formula and check if the result equals the radius, which is 5 units in this case. If the calculated distance is exactly 5, then the point lies on the circle; otherwise, it does not. The distance formula is a cornerstone of coordinate geometry, and its application extends far beyond this particular problem. It is used in various fields, including navigation, computer graphics, and engineering, to calculate distances and determine spatial relationships between points.

Applying the Distance Formula to Option (A): (2, √21)

Let's apply the distance formula to the first point, (2, √21). We substitute x = 2 and y = √21 into the formula √(x² + y²). This gives us √(2² + (√21)²), which simplifies to √(4 + 21) = √25. The square root of 25 is 5, which is exactly the radius of our circle. Therefore, the point (2, √21) is indeed located on the circle. We have found one point that satisfies the condition, but let's continue to examine the other options to ensure we understand the process thoroughly and to see if there might be other points that also lie on the circle. This step-by-step approach is crucial in problem-solving, allowing us to break down complex problems into manageable steps and arrive at the correct solution with confidence. Furthermore, this exercise reinforces our understanding of how the distance formula works and how it can be applied to solve geometric problems.

Analyzing Option (B): (2, √23)

Now, let's analyze option (B), the point (2, √23). We'll use the same distance formula, substituting x = 2 and y = √23 into √(x² + y²). This yields √(2² + (√23)²), which simplifies to √(4 + 23) = √27. The square root of 27 is not equal to 5. In fact, it is greater than 5, since 27 is greater than 25. Therefore, the point (2, √23) does not lie on the circle. This result highlights the importance of precise calculation and comparison. While √23 is close to √21, the difference is enough to place the point (2, √23) outside the circle with a radius of 5. This reinforces the concept that only points at an exact distance of 5 units from the origin will lie on the circle. Analyzing this option not only helps us eliminate it as a possible answer but also deepens our understanding of the geometric constraints of the problem.

Evaluating Option (C): (2, 1)

Moving on to option (C), the point (2, 1), we apply the distance formula again. Substituting x = 2 and y = 1 into √(x² + y²), we get √(2² + 1²), which simplifies to √(4 + 1) = √5. The square root of 5 is significantly less than 5. Therefore, the point (2, 1) does not lie on the circle. This point is much closer to the origin than the radius of the circle, placing it well inside the circle's boundary. This analysis further emphasizes the precise nature of the condition for a point to lie on a circle. The distance from the center must be exactly equal to the radius; any deviation, whether greater or smaller, means the point is not on the circle. Evaluating this option helps to solidify our understanding of the relationship between a point's coordinates, the circle's radius, and the distance formula.

Examining Option (D): (2, 3)

Finally, let's examine option (D), the point (2, 3). Using the distance formula, we substitute x = 2 and y = 3 into √(x² + y²). This gives us √(2² + 3²), which simplifies to √(4 + 9) = √13. The square root of 13 is not equal to 5. It is less than 5, but greater than the result we obtained for option C. Therefore, the point (2, 3) also does not lie on the circle. This point, like option C, is located inside the circle, but it is farther from the origin than (2, 1). This final analysis reinforces the importance of accurate calculations and comparisons. By examining all the options, we have gained a comprehensive understanding of which points satisfy the condition of lying on the circle and which do not. This thorough approach ensures that we have not only found the correct answer but also solidified our understanding of the underlying concepts.

Conclusion: The Point on the Circle

In conclusion, by applying the distance formula to each of the given points, we determined that only point (A), (2, √21), lies on the circle centered at the origin with a radius of 5 units. The distance formula, √((x₂ - x₁)² + (y₂ - y₁)²), is a powerful tool for solving problems in coordinate geometry, allowing us to calculate distances between points and determine their spatial relationships. This exercise has not only helped us identify the correct answer but also deepened our understanding of circles, distances, and the application of the distance formula. The ability to solve such problems is crucial in various fields, from mathematics and physics to engineering and computer science. By mastering these fundamental concepts, we can tackle more complex geometrical challenges with confidence and precision. This thorough exploration has highlighted the importance of meticulous calculation, comparison, and a step-by-step approach to problem-solving.