A Circle Centered at the Origin

Oct 13, 2025 by miodeya.com 13 views

A Circle Centered at the Origin: Exploring the Properties of a Circle

In mathematics, a circle is a set of points that are all equidistant from a central point, known as the center. The distance from the center to any point on the circle is called the radius. In this article, we will explore the properties of a circle centered at the origin and determine whether a given point lies on the circle.

The Equation of a Circle

The equation of a circle centered at the origin is given by:

x^2 + y^2 = r^2

where (x, y) are the coordinates of any point on the circle, and r is the radius of the circle.

A Circle Centered at the Origin

Let's consider a circle centered at the origin, which means the center is at the point (0, 0). The equation of this circle is:

x^2 + y^2 = r^2

Since the circle is centered at the origin, the distance from the center to any point on the circle is equal to the radius.

The Point (0, -9)

We are given that the point (0, -9) lies on the circle. To verify this, we can substitute the coordinates of the point into the equation of the circle:

(0)^2 + (-9)^2 = r^2 0 + 81 = r^2 r^2 = 81

This means that the radius of the circle is √81 = 9.

The Point (8, √17)

Now, let's consider the point (8, √17). We need to determine whether this point lies on the circle. To do this, we can substitute the coordinates of the point into the equation of the circle:

(8)^2 + (√17)^2 = r^2 64 + 17 = r^2 r^2 = 81

This means that the distance from the center to the point (8, √17) is √81 = 9, which is the same as the radius of the circle.

Conclusion

However, we need to consider the coordinates of the point (8, √17) more carefully. The x-coordinate is 8, which is positive, and the y-coordinate is √17, which is positive. This means that the point (8, √17) lies in the first quadrant of the coordinate plane.

The Distance from the Center to the Point

The distance from the center to the point (8, √17) can be calculated using the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) is the center of the circle (0, 0), and (x2, y2) is the point (8, √17).

d = √((8 - 0)^2 + (√17 - 0)^2) d = √(64 + 17) d = √81 d = 9

This means that the distance from the center to the point (8, √17) is 9, which is the same as the radius of the circle.

The Final Answer

However, we need to consider the fact that the point (8, √17) lies in the first quadrant of the coordinate plane, and the circle is centered at the origin. This means that the point (8, √17) lies on the circle, but it is not the only point on the circle that lies in the first quadrant.

The Answer to the Question

To answer the question, we need to consider the fact that the point (8, √17) lies on the circle, but it is not the only point on the circle that lies in the first quadrant. Therefore, the answer to the question is:

Yes, the point (8, √17) does lie on the circle.

Explanation

The point (8, √17) lies on the circle because the distance from the center to the point is equal to the radius of the circle. The point (8, √17) lies in the first quadrant of the coordinate plane, and the circle is centered at the origin. Therefore, the point (8, √17) lies on the circle.

Conclusion

In conclusion, the point (8, √17) does lie on the circle. The distance from the center to the point is equal to the radius of the circle, and the point lies in the first quadrant of the coordinate plane. Therefore, the answer to the question is yes, the point (8, √17) does lie on the circle.
A Circle Centered at the Origin: Exploring the Properties of a Circle

Q: What is the equation of a circle centered at the origin?

A: The equation of a circle centered at the origin is given by:

x^2 + y^2 = r^2

where (x, y) are the coordinates of any point on the circle, and r is the radius of the circle.

Q: What is the significance of the radius in a circle?

A: The radius of a circle is the distance from the center to any point on the circle. It is a fundamental property of a circle and is used to determine the size and shape of the circle.

Q: How do you determine if a point lies on a circle?

A: To determine if a point lies on a circle, you can substitute the coordinates of the point into the equation of the circle. If the equation is true, then the point lies on the circle.

Q: What is the distance formula used to calculate the distance between two points?

A: The distance formula is used to calculate the distance between two points (x1, y1) and (x2, y2) in a coordinate plane. The formula is:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Q: How do you calculate the distance from the center to a point on a circle?

A: To calculate the distance from the center to a point on a circle, you can use the distance formula. If the point lies on the circle, then the distance from the center to the point is equal to the radius of the circle.

Q: What is the significance of the first quadrant in a coordinate plane?

A: The first quadrant in a coordinate plane is the region where both the x and y coordinates are positive. It is one of the four quadrants in a coordinate plane and is used to determine the location of points in the plane.

Q: How do you determine if a point lies in the first quadrant?

A: To determine if a point lies in the first quadrant, you can check if both the x and y coordinates are positive. If both coordinates are positive, then the point lies in the first quadrant.

Q: What is the relationship between the point (8, √17) and the circle centered at the origin?

A: The point (8, √17) lies on the circle centered at the origin because the distance from the center to the point is equal to the radius of the circle.

Q: Why is the point (8, √17) significant in the context of the circle centered at the origin?

A: The point (8, √17) is significant because it lies on the circle centered at the origin and is used to illustrate the properties of a circle.

Q: What is the final answer to the question of whether the point (8, √17) lies on the circle centered at the origin?

A: The final answer to the question is yes, the point (8, √17) does lie on the circle centered at the origin.

Conclusion

In conclusion, the point (8, √17) does lie on the circle centered at the origin. The distance from the center to the point is equal to the radius of the circle, and the point lies in the first quadrant of the coordinate plane. Therefore, the answer to the question is yes, the point (8, √17) does lie on the circle centered at the origin.