Points On A Circle Centered At The Origin A Detailed Explanation

by THE IDEN 65 views

Determining which point lies on a circle is a fundamental concept in coordinate geometry. This article will delve into the method of finding such points, particularly for a circle centered at the origin. We'll use the distance formula, a crucial tool in this endeavor, and apply it to a specific problem: identifying which point, among the given options, lies on a circle centered at the origin with a radius of 5 units.

Understanding the Equation of a Circle and the Distance Formula

At the heart of this problem lies the equation of a circle. A circle centered at the origin (0, 0) with a radius r has the equation:

x² + y² = r²

This equation stems directly from the distance formula, which calculates the distance between two points in a coordinate plane. The distance formula is given by:

√[( x₂ - x₁)² + (y₂ - y₁)²]

In the context of a circle centered at the origin, one point is always (0, 0). If we let ( x, y) be any point on the circle, the distance between (0, 0) and (x, y) must equal the radius r. Applying the distance formula, we get:

√[( x - 0)² + (y - 0)²] = r

Squaring both sides, we arrive at the familiar equation of a circle centered at the origin:

x² + y² = r²

This equation tells us that any point (x, y) that satisfies this equation lies on the circle. Conversely, any point that does not satisfy this equation does not lie on the circle. To determine if a point lies on a given circle, substitute the x and y coordinates of the point into the equation of the circle. If the equation holds true, the point lies on the circle; otherwise, it does not.

In our specific problem, we have a circle centered at the origin with a radius of 5 units. Therefore, the equation of our circle is:

x² + y² = 5²

x² + y² = 25

Our task is to test each of the given points to see which one satisfies this equation.

Applying the Distance Formula to the Problem

Now, let's apply this understanding to the specific problem. We are given four points:

A. (2, √21) B. (2, √23) C. (2, 1) D. (2, 3)

We need to determine which of these points lies on the circle centered at the origin with a radius of 5 units. To do this, we will substitute the x and y coordinates of each point into the equation x² + y² = 25 and see if the equation holds true.

Testing Point A (2, √21)

Substitute x = 2 and y = √21 into the equation:

2² + (√21)² = 4 + 21 = 25

Since 25 = 25, Point A (2, √21) lies on the circle.

Testing Point B (2, √23)

Substitute x = 2 and y = √23 into the equation:

2² + (√23)² = 4 + 23 = 27

Since 27 ≠ 25, Point B (2, √23) does not lie on the circle.

Testing Point C (2, 1)

Substitute x = 2 and y = 1 into the equation:

2² + 1² = 4 + 1 = 5

Since 5 ≠ 25, Point C (2, 1) does not lie on the circle.

Testing Point D (2, 3)

Substitute x = 2 and y = 3 into the equation:

2² + 3² = 4 + 9 = 13

Since 13 ≠ 25, Point D (2, 3) does not lie on the circle.

Conclusion: Identifying the Point on the Circle

After testing each point, we found that only Point A (2, √21) satisfies the equation x² + y² = 25. Therefore, Point A (2, √21) is the only point among the given options that lies on the circle centered at the origin with a radius of 5 units.

This problem highlights the importance of understanding the relationship between the equation of a circle and the distance formula. By applying these concepts, we can effectively determine whether a given point lies on a circle or not. This fundamental skill is crucial in various areas of mathematics and physics, particularly in geometry and coordinate systems.

Further Exploration of Circles and the Distance Formula

The concept of circles and the distance formula extends beyond simple point verification. Understanding these concepts opens doors to more complex problems, such as finding the equation of a circle given certain conditions, determining the intersection points of circles and lines, and exploring geometric transformations involving circles. Further exploration of these topics will deepen your understanding of coordinate geometry and its applications.

Applications in Real-World Scenarios

The principles discussed here have real-world applications in various fields, including:

  • Navigation: GPS systems rely on the distance formula to calculate distances between satellites and receivers, determining the receiver's location.
  • Computer Graphics: Circles and distances are fundamental in creating and manipulating images and animations.
  • Engineering: Designing circular structures, such as arches and domes, requires a thorough understanding of circle geometry.
  • Astronomy: Calculating orbits of celestial bodies involves understanding distances and circular paths.

Expanding Your Knowledge

To further expand your knowledge, consider exploring the following topics:

  • General Equation of a Circle: Learn about the equation of a circle with a center at any point (h, k).
  • Tangents and Secants: Investigate the properties of lines that touch a circle at one point (tangents) or intersect it at two points (secants).
  • Conic Sections: Discover how circles relate to other conic sections, such as ellipses, parabolas, and hyperbolas.
  • Analytic Geometry: Delve deeper into the connection between algebra and geometry, using algebraic techniques to solve geometric problems.

By mastering these concepts, you'll gain a solid foundation in mathematics and its practical applications. The ability to determine points on a circle, derived from the distance formula, is a cornerstone of geometric understanding and problem-solving.