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

To determine which point lies on a circle centered at the origin with a radius of 5 units, we need to understand the equation of a circle and how to apply the distance formula. This article will delve into the concepts, provide a step-by-step solution, and offer insights into related mathematical principles.

Understanding the Equation of a Circle

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 is derived from the Pythagorean theorem and represents all points (x, y) that are a distance r away from the origin. For our specific problem, the radius r is 5 units, so the equation becomes:

x² + y² = 5² x² + y² = 25

This equation is our key to verifying whether a given point lies on the circle. A point (x, y) lies on the circle if and only if its coordinates satisfy this equation. In simpler terms, if we plug in the x and y values of a point into the equation, and the equation holds true (i.e., the left side equals the right side), then the point is on the circle. If the equation doesn't hold true, the point is either inside or outside the circle.

To truly grasp this concept, it's beneficial to understand its connection to the distance formula. The distance formula, derived from the Pythagorean theorem, calculates the distance between two points in a coordinate plane. Given two points (x₁, y₁) and (x₂, y₂), the distance d between them is:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

In the context of a circle centered at the origin, we can consider one point as the origin (0, 0) and the other point as any point (x, y) on the circle. The distance d then becomes the radius r. Applying the distance formula, we get:

r = √[(x - 0)² + (y - 0)²] r = √(x² + y²)

Squaring both sides, we arrive back at the equation of the circle:

r² = x² + y²

This connection highlights that the equation of a circle is essentially a restatement of the distance formula in a specific context – the distance from any point on the circle to the center is constant and equal to the radius. Understanding this relationship provides a deeper appreciation for the geometry of circles and their algebraic representation.

Applying the Distance Formula

The distance formula, as provided, is:

√[( x₂ - x₁ )² + ( y₂ - y₁ )²]

This formula calculates the distance between two points in a coordinate plane. In our problem, we're dealing with a circle centered at the origin (0, 0). Therefore, one of our points will always be (0, 0). The other point will be one of the options provided (A, B, C, or D). We want to find the point that is exactly 5 units away from the origin, which is the radius of our circle.

To apply the formula effectively, let's break it down step-by-step. We'll denote the coordinates of the point we're testing as (x, y). The distance d between (x, y) and the origin (0, 0) is:

d = √[( x - 0 )² + ( y - 0 )²] d = √(x² + y²)

Now, we want to find the point where this distance d is equal to the radius, which is 5. So, we set d = 5:

5 = √(x² + y²)

To get rid of the square root, we square both sides of the equation:

25 = x² + y²

This is the same equation we derived earlier for a circle centered at the origin with a radius of 5. This equation provides a direct method for testing whether a point lies on the circle. We simply plug in the x and y coordinates of the point into the equation. If the equation holds true (i.e., the left side equals the right side), then the point is on the circle. If it doesn't hold true, the point is not on the circle.

For each option (A, B, C, and D), we will substitute the x and y values into this equation and see if the result is 25. This method is more efficient than directly calculating the distance using the distance formula, as it avoids the square root calculation until the very end. It's a prime example of how understanding the underlying principles of geometry and algebra can lead to more streamlined problem-solving approaches.

Step-by-Step Solution

Now, let's apply this knowledge to the given options. We will test each point to see if it satisfies the equation x² + y² = 25.

A. (2, √21)

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

2² + (√21)² = 4 + 21 = 25

Since the equation holds true, the point (2, √21) lies on the circle.

B. (2, √23)

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

2² + (√23)² = 4 + 23 = 27

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

C. (2, 1)

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

2² + 1² = 4 + 1 = 5

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

D. (2, 3)

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

2² + 3² = 4 + 9 = 13

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

Therefore, after testing all the options, we find that only point A (2, √21) satisfies the equation of the circle. This confirms that point A is the only point among the given options that lies on the circle centered at the origin with a radius of 5 units.

This systematic approach of substituting the coordinates into the circle's equation is a reliable method for determining whether a point lies on a circle. It leverages the fundamental relationship between the circle's equation, the distance formula, and the Pythagorean theorem.

Conclusion

In conclusion, to determine which point lies on a circle centered at the origin with a given radius, we use the equation of the circle x² + y² = r². By substituting the coordinates of each point into the equation, we can verify if the equation holds true. In this case, only point A (2, √21) satisfies the equation for a circle with a radius of 5 units. Understanding the relationship between the equation of a circle and the distance formula is crucial for solving such problems efficiently. This method provides a solid foundation for tackling more complex geometric problems involving circles and coordinate geometry.