Identifying Points On A Circle Centered At The Origin
Determining whether a point lies on a circle is a fundamental concept in coordinate geometry. In this article, we will explore how to identify points on a circle centered at the origin with a radius of 5 units. We'll utilize the distance formula, a cornerstone of coordinate geometry, to verify which of the given points satisfy the circle's equation. This exploration will not only solidify your understanding of circles and distances but also enhance your problem-solving skills in analytical geometry.
Understanding the Circle's Equation and the Distance Formula
To effectively identify points on a circle, it's crucial to grasp the circle's equation and the distance formula. A circle centered at the origin (0, 0) with a radius r has the equation x² + y² = r². This equation stems from the Pythagorean theorem and represents all points (x, y) that are a distance r away from the origin. The distance formula, derived from the Pythagorean theorem, is given by:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
This formula calculates the distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane. In our case, one point is always the origin (0, 0), and the other point is the candidate point we want to test. By applying the distance formula and comparing the result to the radius, we can determine if a point lies on the circle.
Applying the Distance Formula to Identify Points on the Circle
In this specific scenario, we are given a circle centered at the origin with a radius of 5 units. This means that any point (x, y) lying on the circle must satisfy the equation x² + y² = 5², which simplifies to x² + y² = 25. We are provided with four potential points: A. (2, √21), B. (2, √23), C. (2, 1), and D. (2, 3). To determine which of these points lies on the circle, we will calculate the distance of each point from the origin using the distance formula and check if it equals the radius, which is 5.
For point A (2, √21), the distance from the origin is:
Distance = √[(2 - 0)² + (√21 - 0)²] = √(4 + 21) = √25 = 5
Since the distance is equal to the radius, point A (2, √21) lies on the circle. For point B (2, √23), the distance from the origin is:
Distance = √[(2 - 0)² + (√23 - 0)²] = √(4 + 23) = √27 ≈ 5.196
The distance for point B is approximately 5.196, which is not equal to the radius of 5. Therefore, point B does not lie on the circle. For point C (2, 1), the distance from the origin is:
Distance = √[(2 - 0)² + (1 - 0)²] = √(4 + 1) = √5 ≈ 2.236
The distance for point C is approximately 2.236, which is significantly less than the radius of 5. Thus, point C does not lie on the circle. For point D (2, 3), the distance from the origin is:
Distance = √[(2 - 0)² + (3 - 0)²] = √(4 + 9) = √13 ≈ 3.606
The distance for point D is approximately 3.606, which is also less than the radius of 5. Consequently, point D does not lie on the circle. Therefore, after analyzing all the points, we can conclude that only point A (2, √21) lies on the circle centered at the origin with a radius of 5 units.
Step-by-Step Verification of Points
Let's delve deeper into the step-by-step verification process for each point. This will provide a clearer understanding of how we arrived at the conclusion that only point A lies on the circle. The process involves substituting the coordinates of each point into the distance formula and comparing the calculated distance with the radius of the circle.
Point A (2, √21)
To verify if point A (2, √21) lies on the circle, we substitute its coordinates into the distance formula:
Distance = √[(2 - 0)² + (√21 - 0)²]
First, we calculate the squares of the differences:
(2 - 0)² = 2² = 4
(√21 - 0)² = (√21)² = 21
Next, we add these values together:
4 + 21 = 25
Finally, we take the square root of the sum:
√25 = 5
The distance of point A from the origin is 5 units, which is equal to the radius of the circle. Therefore, point A (2, √21) lies on the circle.
Point B (2, √23)
For point B (2, √23), we follow the same procedure:
Distance = √[(2 - 0)² + (√23 - 0)²]
Calculate the squares of the differences:
(2 - 0)² = 2² = 4
(√23 - 0)² = (√23)² = 23
Add the values:
4 + 23 = 27
Take the square root of the sum:
√27 ≈ 5.196
The distance of point B from the origin is approximately 5.196 units, which is not equal to the radius of 5. Therefore, point B (2, √23) does not lie on the circle.
Point C (2, 1)
For point C (2, 1), we apply the distance formula:
Distance = √[(2 - 0)² + (1 - 0)²]
Calculate the squares of the differences:
(2 - 0)² = 2² = 4
(1 - 0)² = 1² = 1
Add the values:
4 + 1 = 5
Take the square root of the sum:
√5 ≈ 2.236
The distance of point C from the origin is approximately 2.236 units, which is significantly less than the radius of 5. Therefore, point C (2, 1) does not lie on the circle.
Point D (2, 3)
Finally, for point D (2, 3):
Distance = √[(2 - 0)² + (3 - 0)²]
Calculate the squares of the differences:
(2 - 0)² = 2² = 4
(3 - 0)² = 3² = 9
Add the values:
4 + 9 = 13
Take the square root of the sum:
√13 ≈ 3.606
The distance of point D from the origin is approximately 3.606 units, which is also less than the radius of 5. Therefore, point D (2, 3) does not lie on the circle.
Conclusion
In conclusion, by applying the distance formula to each point and comparing the result with the radius of the circle, we have successfully identified that only point A (2, √21) lies on the circle centered at the origin with a radius of 5 units. This exercise demonstrates the practical application of the distance formula in determining the position of points relative to a circle. Understanding these concepts is crucial for further exploration in geometry and related fields. The ability to accurately calculate distances and apply geometric principles is a valuable skill that extends beyond academic settings, finding applications in various real-world scenarios such as navigation, engineering, and computer graphics. Therefore, mastering these fundamental concepts is an investment in your problem-solving abilities and analytical thinking.
Practice Problems
To solidify your understanding of finding points on a circle, try these practice problems:
- Which point is on the circle centered at the origin with a radius of 10 units: (6, 8), (5, 5), (7, 7), or (8, 4)?
- Determine if the point (3, √7) lies on the circle x² + y² = 16.
- Find a point on the circle centered at the origin with a radius of 13 units, given that the x-coordinate is 5.
Working through these problems will help reinforce your grasp of the concepts discussed in this article. Remember to use the distance formula and the circle's equation to verify your answers. Happy solving!