Identifying Points On A Circle Exploring Coordinate Geometry And The Distance Formula

Determining which point lies on a circle centered at the origin with a specific radius is a fundamental concept in coordinate geometry. This article dives deep into understanding circles, their equations, and how to apply the distance formula to solve such problems. We'll break down the key concepts, explore the relevant formulas, and then apply them to the given options to identify the point that resides on the circle with a radius of 5 units centered at the origin. Let's embark on this geometrical journey!

Understanding the Equation of a Circle

The first step in solving this problem is to grasp the equation of a circle. A circle is defined as the set of all points equidistant from a central point. This fixed distance is known as the radius of the circle. When the circle is centered at the origin (0, 0) in a Cartesian coordinate system, its equation takes a simplified form, making calculations more manageable. The equation of a circle centered at the origin (0, 0) with a radius 'r' is given by:

$$x^2 + y^2 = r^2$$

This equation is derived directly from the Pythagorean theorem and represents the relationship between the x and y coordinates of any point on the circle and the circle's radius. In our case, the circle is centered at the origin and has a radius of 5 units. Therefore, we can substitute 'r' with 5 in the equation:

$$x^2 + y^2 = 5^2$$

$$x^2 + y^2 = 25$$

This equation, x^2 + y^2 = 25, is the specific equation for the circle we are dealing with in this problem. Any point (x, y) that satisfies this equation lies on the circle. To determine which of the given points lies on the circle, we need to substitute the x and y coordinates of each point into this equation and see if the equation holds true. This is where the beauty of coordinate geometry shines – we can use algebraic equations to represent and analyze geometric shapes.

The significance of this equation lies in its ability to provide a direct test for whether a point lies on the circle. If the sum of the squares of the x and y coordinates of a point equals the square of the radius, then that point is a member of the circle's infinite set of points. This principle will guide us as we evaluate the given options. We'll be looking for the pair of coordinates that, when plugged into the equation, yields the value 25. This is a powerful technique, not just for circles centered at the origin, but as a foundation for understanding more complex geometric problems involving circles in various positions and orientations.

Applying the Distance Formula

While the equation of a circle provides a direct method for checking if a point lies on the circle, the distance formula offers an alternative approach that underscores the fundamental definition of a circle. The distance formula calculates the distance between two points in a coordinate plane, and in this context, it allows us to determine the distance between the origin (the center of the circle) and each of the given points. If this distance is equal to the radius of the circle (5 units), then the point lies on the circle. The distance formula is derived from the Pythagorean theorem and is expressed as:

$$Distance = \sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$$

Where (x1, y1) and (x2, y2) are the coordinates of the two points. In our case, one point is always the origin (0, 0), and the other point is one of the options given (e.g., (2, √21)). We will calculate the distance between the origin and each point and compare it to the radius, 5 units.

The distance formula is a versatile tool in coordinate geometry. It's not just limited to circle problems; it can be used to find the distance between any two points, determine the lengths of line segments, and even prove geometric theorems. Understanding the distance formula is crucial for solving a wide range of problems in geometry and related fields. By applying the distance formula in this context, we are essentially verifying if the point satisfies the core definition of a circle – being equidistant from the center. For a point to lie on the circle, its distance from the origin must be exactly 5 units. Any distance greater or less than 5 would place the point outside or inside the circle, respectively.

Using the distance formula reinforces the geometric interpretation of the circle's equation. The equation x^2 + y^2 = r^2 is, in essence, a restatement of the Pythagorean theorem applied to the radius and the coordinates of a point on the circle. The distance formula provides a more intuitive understanding of this relationship, as it directly calculates the distance and compares it to the radius. This dual approach – using the equation and the distance formula – not only helps solve the problem but also deepens our understanding of the underlying geometric principles.

Evaluating the Options

Now, let's put our knowledge into practice and evaluate each of the given options using both the equation of the circle and the distance formula. This will not only help us find the correct answer but also solidify our understanding of the concepts involved. We'll systematically substitute the coordinates of each point into the equation x^2 + y^2 = 25 and calculate the distance between each point and the origin using the distance formula. By comparing the results, we can definitively determine which point lies on the circle.

Option A: (2, √21)

• Using the equation of the circle: Substitute x = 2 and y = √21 into the equation x^2 + y^2 = 25: 2^2 + (√21)^2 = 4 + 21 = 25. The equation holds true.
• Using the distance formula: Calculate the distance between (2, √21) and the origin (0, 0): Distance = √((2-0)^2 + (√21-0)^2) = √(4 + 21) = √25 = 5. The distance is equal to the radius.

Both methods confirm that the point (2, √21) lies on the circle.

Option B: (2, √23)

• Using the equation of the circle: Substitute x = 2 and y = √23 into the equation x^2 + y^2 = 25: 2^2 + (√23)^2 = 4 + 23 = 27. The equation does not hold true.
• Using the distance formula: Calculate the distance between (2, √23) and the origin (0, 0): Distance = √((2-0)^2 + (√23-0)^2) = √(4 + 23) = √27 ≈ 5.2. The distance is not equal to the radius.

Both methods indicate that the point (2, √23) does not lie on the circle.

Option C: (2, 1)

• Using the equation of the circle: Substitute x = 2 and y = 1 into the equation x^2 + y^2 = 25: 2^2 + 1^2 = 4 + 1 = 5. The equation does not hold true.
• Using the distance formula: Calculate the distance between (2, 1) and the origin (0, 0): Distance = √((2-0)^2 + (1-0)^2) = √(4 + 1) = √5 ≈ 2.24. The distance is not equal to the radius.

Both methods clearly show that the point (2, 1) does not lie on the circle.

Option D: (2, 3)

• Using the equation of the circle: Substitute x = 2 and y = 3 into the equation x^2 + y^2 = 25: 2^2 + 3^2 = 4 + 9 = 13. The equation does not hold true.
• Using the distance formula: Calculate the distance between (2, 3) and the origin (0, 0): Distance = √((2-0)^2 + (3-0)^2) = √(4 + 9) = √13 ≈ 3.61. The distance is not equal to the radius.

Again, both methods demonstrate that the point (2, 3) does not lie on the circle.

Through this systematic evaluation, we can confidently conclude that only option A, the point (2, √21), satisfies the conditions to lie on the circle centered at the origin with a radius of 5 units. This comprehensive analysis highlights the power of both the circle's equation and the distance formula in solving geometric problems.

Conclusion

In conclusion, we have successfully identified the point that lies on the circle centered at the origin with a radius of 5 units. By understanding the equation of a circle and the distance formula, we were able to systematically evaluate each option and determine that (2, √21) is the correct answer. This exercise demonstrates the importance of fundamental concepts in coordinate geometry and their application in solving practical problems. The combination of algebraic equations and geometric intuition provides a powerful approach to understanding and analyzing shapes and their properties in a coordinate plane.