Finding Points On A Circle Centered At Origin With Radius 5

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Determining whether a point lies on a circle is a fundamental concept in geometry. In this comprehensive guide, we will delve into the method of finding points on a circle centered at the origin with a radius of 5 units. We'll explore the distance formula, its application in this scenario, and meticulously analyze each given option to pinpoint the correct answer. This exploration is crucial for anyone studying analytical geometry, as it combines algebraic equations with geometric concepts.

Understanding the Distance Formula and Circles

At the heart of this problem lies the distance formula, a cornerstone of coordinate geometry. The distance formula is mathematically expressed as:

(x2−x1)2+(y2−y1)2\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}

This formula calculates the straight-line distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in a coordinate plane. Now, let's connect this to the concept of a circle. A circle is defined as the set of all points equidistant from a central point. This constant distance is the radius of the circle. When the circle is centered at the origin (0, 0), the distance formula simplifies our task of finding points on the circle. For a circle centered at the origin with a radius r, any point (x, y) on the circle must satisfy the equation:

x2+y2=r2x^2 + y^2 = r^2

This equation is derived directly from the distance formula, where the distance between the origin (0, 0) and any point (x, y) on the circle is equal to the radius r. In our specific problem, the radius r is given as 5 units. Therefore, any point (x, y) that lies on the circle centered at the origin with a radius of 5 must satisfy the equation:

x2+y2=52x^2 + y^2 = 5^2

Which simplifies to:

x2+y2=25x^2 + y^2 = 25

This equation forms the basis for our analysis. We will now substitute the coordinates of each given point into this equation to check if it holds true. If the equation is satisfied, the point lies on the circle; otherwise, it does not. This method provides a straightforward and accurate way to determine the position of a point relative to a circle centered at the origin. Understanding this principle is not only vital for solving this particular problem but also for tackling various other problems in coordinate geometry, such as finding the equation of a circle, determining the intersection of a line and a circle, and more. The distance formula and its application in defining circles form a fundamental building block in the study of geometric shapes in the coordinate plane.

Analyzing the Options

Now that we've established the equation x2+y2=25x^2 + y^2 = 25, we can methodically analyze each option to determine which point lies on the circle. This process involves substituting the x and y coordinates of each point into the equation and verifying if the equality holds true. It's a direct application of the geometric principle we discussed earlier, where we use algebraic methods to confirm the location of points on a circle. This step-by-step analysis is crucial for accuracy and understanding the underlying concepts.

Option A: (2,21)(2, \sqrt{21})

For option A, the point is (2,21)(2, \sqrt{21}). Let's substitute these values into our equation:

22+(21)2=4+21=252^2 + (\sqrt{21})^2 = 4 + 21 = 25

Since the equation holds true, this point lies on the circle. This demonstrates the power of using the equation of a circle to verify the position of a point. By substituting the coordinates and performing the calculations, we can definitively determine if a point is part of the circle's circumference. This method is not only efficient but also provides a clear and logical approach to solving geometric problems. The fact that this point satisfies the equation confirms that it is located exactly 5 units away from the origin, which is the defining characteristic of a point on a circle with a radius of 5 centered at the origin.

Option B: (2,23)(2, \sqrt{23})

Moving on to option B, the point is (2,23)(2, \sqrt{23}). Substituting these values into the equation, we get:

22+(23)2=4+23=272^2 + (\sqrt{23})^2 = 4 + 23 = 27

In this case, the sum is 27, which is not equal to 25. Therefore, this point does not lie on the circle. This result highlights the importance of precise calculations in coordinate geometry. Even a small difference in the sum can indicate that a point is not located on the circle. This is because the equation x2+y2=25x^2 + y^2 = 25 defines the circle's boundary with strict precision. Any point that deviates from this equation, even slightly, will fall either inside or outside the circle. This underscores the sensitivity of the equation to the coordinates of the point and the accuracy required in determining the points that lie on the circle.

Option C: (2,1)(2, 1)

For option C, the point is (2, 1). Substituting these values into the equation, we have:

22+12=4+1=52^2 + 1^2 = 4 + 1 = 5

The sum is 5, which is significantly less than 25. Hence, this point is not on the circle. This result illustrates that the point (2, 1) is much closer to the origin than a point on the circle with a radius of 5 would be. The equation x2+y2=25x^2 + y^2 = 25 serves as a quantitative measure of the distance of a point from the origin. A smaller sum indicates that the point is closer to the origin, while a larger sum would indicate that the point is farther away. In this case, the sum being 5 clearly shows that the point (2, 1) is located well within the circle's boundary, emphasizing the crucial role of the radius in defining the circle's extent.

Option D: (2,3)(2, 3)

Lastly, let's analyze option D, the point (2, 3). Substituting these values into the equation, we get:

22+32=4+9=132^2 + 3^2 = 4 + 9 = 13

Again, the sum is 13, which is not equal to 25. Therefore, this point does not lie on the circle. Similar to option C, this result indicates that the point (2, 3) is also located inside the circle, but further away from the origin compared to the point (2, 1). The difference in the sums (13 compared to 5) reflects the relative distances of these points from the origin. The point (2, 3) is farther from the origin than (2, 1) but still not far enough to be on the circle with a radius of 5. This analysis reinforces the concept that the equation x2+y2=25x^2 + y^2 = 25 precisely defines the boundary of the circle, and only points that satisfy this equation will lie on its circumference.

Conclusion

Through our detailed analysis, we've determined that option A, the point (2,21)(2, \sqrt{21}), is the only point that lies on the circle centered at the origin with a radius of 5 units. This conclusion was reached by applying the distance formula and the equation of a circle, which are fundamental tools in coordinate geometry. Understanding these concepts and their application is essential for solving a wide range of problems involving circles, distances, and geometric shapes in the coordinate plane. The step-by-step approach we used, involving substituting coordinates into the equation and verifying the equality, provides a clear and effective method for determining the position of points relative to a circle. This process not only helps in solving specific problems but also enhances our understanding of the relationship between algebraic equations and geometric figures.

Therefore, the correct answer is A. (2,21)(2, \sqrt{21}).

Repair Input Keyword

Which of the following points is located on a circle centered at the origin with a radius of 5 units?