Circle At Origin Does (8, √17) Lie On It A Geometric Exploration

In the fascinating realm of geometry, circles hold a special place. A circle, defined as the set of all points equidistant from a central point, exhibits a beautiful symmetry and is fundamental to many mathematical concepts. When a circle is centered at the origin (0,0) of a coordinate plane, its properties can be elegantly described using the distance formula, which stems directly from the Pythagorean theorem. This article delves into the specifics of a circle centered at the origin that contains the point (0,-9), and we'll investigate whether the point (8, √17) also lies on this circle. This exploration will not only reinforce our understanding of circles but also demonstrate the practical application of geometric principles in coordinate geometry.

Determining the Radius of the Circle

To begin our investigation, we must first determine the radius of the circle. In geometry, the radius is the distance from the center of the circle to any point on its circumference. Since we know the circle is centered at the origin (0,0) and contains the point (0,-9), we can use the distance formula to calculate the radius. The distance formula, derived from the Pythagorean theorem, is given by:

√[(x₂ - x₁)² + (y₂ - y₁)²]

Where (x₁, y₁) and (x₂, y₂) are the coordinates of two points. In our case, (x₁, y₁) = (0,0) (the center of the circle) and (x₂, y₂) = (0,-9) (a point on the circle). Plugging these values into the distance formula, we get:

√[(0 - 0)² + (-9 - 0)²] = √(0² + (-9)²) = √81 = 9

Therefore, the radius of the circle is 9 units. This means that any point on the circle must be exactly 9 units away from the origin. Understanding this foundational concept is crucial for determining whether other points, such as (8, √17), lie on the circle. The radius serves as the defining characteristic of our circle, and it will be the benchmark against which we measure the distance of any other potential point on the circumference. This initial calculation of the radius sets the stage for the subsequent analysis of other points and their relationship to the circle.

Verifying if (8, √17) Lies on the Circle

Now that we have established the radius of the circle as 9 units, our next step is to determine whether the point (8, √17) lies on this circle. To do this, we will again employ the distance formula, this time calculating the distance between the origin (0,0) and the point (8, √17). If this distance is equal to the radius (9 units), then the point (8, √17) lies on the circle. If it is not, then the point lies either inside or outside the circle.

Applying the distance formula with (x₁, y₁) = (0,0) and (x₂, y₂) = (8, √17), we get:

√[(8 - 0)² + (√17 - 0)²] = √(8² + (√17)²) = √(64 + 17) = √81 = 9

The calculation reveals that the distance between the origin and the point (8, √17) is exactly 9 units, which is equal to the radius of the circle. This result confirms that the point (8, √17) does indeed lie on the circle. The precise alignment of this point's distance from the center with the circle's radius underscores the fundamental definition of a circle and reinforces our understanding of how points are positioned relative to a circle in a coordinate plane. This verification step is crucial in solidifying our comprehension of circles and their properties.

Alternative Approach: Using the Equation of a Circle

An alternative and equally insightful method to determine if the point (8, √17) lies on the circle involves using the equation of a circle centered at the origin. The general equation for a circle centered at the origin (0,0) with radius r is given by:

x² + y² = r²

In our case, we have already established that the radius r is 9 units. Therefore, the equation of our circle is:

x² + y² = 9² or x² + y² = 81

To check if the point (8, √17) lies on this circle, we can substitute the x and y coordinates of the point into the equation. If the equation holds true, then the point lies on the circle. Substituting x = 8 and y = √17 into the equation, we get:

8² + (√17)² = 64 + 17 = 81

The result, 81, is equal to r², which means that the equation is satisfied. This confirms that the point (8, √17) lies on the circle. This method provides a powerful algebraic approach to verifying the geometric properties of circles. By leveraging the equation of a circle, we can directly test whether a point's coordinates fit the defining relationship of the circle, reinforcing the connection between algebra and geometry. This alternative approach not only validates our previous findings but also enhances our problem-solving toolkit.

Conclusion

In summary, we have explored the properties of a circle centered at the origin that contains the point (0,-9). By calculating the distance between the origin and this point, we determined the radius of the circle to be 9 units. Subsequently, we investigated whether the point (8, √17) also lies on the circle. Using both the distance formula and the equation of a circle, we conclusively demonstrated that the point (8, √17) does indeed lie on the circle. This exploration has not only reinforced our understanding of circles and their properties but has also highlighted the utility of the distance formula and the equation of a circle in solving geometric problems. The convergence of these two methods in confirming the point's location underscores the robust nature of geometric principles and their application in coordinate geometry. This exercise exemplifies how a solid grasp of fundamental geometric concepts can empower us to solve complex problems with clarity and precision.

Original Question: A circle centered at the origin contains the point (0,-9). Does (8, √17) also lie on the circle?

Repaired Question: Given a circle centered at the origin that passes through the point (0, -9), determine whether the point (8, √17) also lies on the same circle. Explain your reasoning and provide mathematical justification.

Circle at Origin Does (8, √17) Lie on It? A Geometric Exploration