The Given Circle and Point

Oct 13, 2025 by magicalceylontours.com 96 views

Introduction

In geometry, a circle is a set of points that are all equidistant from a central point called the center. The distance from the center to any point on the circle is called the radius. In this article, we will explore a circle centered at the origin and a point that lies on the circle. We will then determine which of the given points also lies on the circle.

The Given Circle and Point

The circle is centered at the origin, which means its center is at the point (0, 0). The point Q(10, 0) lies on the circle. This information gives us a starting point to work with.

Equation of a Circle

The general equation of a circle centered at the origin is x^2 + y^2 = r^2, where r is the radius of the circle. Since the circle is centered at the origin, the equation of the circle is x^2 + y^2 = r^2.

Finding the Radius

We know that the point Q(10, 0) lies on the circle. We can use this information to find the radius of the circle. Since the point Q(10, 0) lies on the circle, it must satisfy the equation of the circle. Plugging in the values of x and y, we get:

10^2 + 0^2 = r^2 100 = r^2 r = √100 r = 10

The Radius of the Circle

The radius of the circle is 10 units.

Which Point Lies on the Circle?

We are given three points: A(5, 5√3), B(4, 5√3), and C. We need to determine which of these points lies on the circle.

Point A

To determine if point A lies on the circle, we need to check if it satisfies the equation of the circle. Plugging in the values of x and y, we get:

5^2 + (5√3)^2 = r^2 25 + 75 = r^2 100 = r^2 r = √100 r = 10

Since the radius of the circle is 10 units, and the distance from the center to point A is also 10 units, point A lies on the circle.

Point B

To determine if point B lies on the circle, we need to check if it satisfies the equation of the circle. Plugging in the values of x and y, we get:

4^2 + (5√3)^2 = r^2 16 + 75 = r^2 91 = r^2 r = √91 r ≠ 10

Since the radius of the circle is 10 units, and the distance from the center to point B is not 10 units, point B does not lie on the circle.

Conclusion

In conclusion, the point A(5, 5√3) lies on the circle, while the point B(4, 5√3) does not lie on the circle. The radius of the circle is 10 units, and the distance from the center to point A is also 10 units.

Key Takeaways

The equation of a circle centered at the origin is x^2 + y^2 = r^2.
The radius of the circle can be found using the distance formula.
To determine if a point lies on a circle, we need to check if it satisfies the equation of the circle.

Final Answer

Q&A: Circle C

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

A: The equation of a circle centered at the origin is x^2 + y^2 = r^2, where r is the radius of the circle.

Q: How do you find the radius of a circle?

A: To find the radius of a circle, you can use the distance formula. If you know the coordinates of a point on the circle, you can plug those values into the equation of the circle to find the radius.

Q: What is the significance of the center of a circle?

A: The center of a circle is the point from which all distances to points on the circle are measured. It is also the point around which the circle is centered.

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

A: To determine if a point lies on a circle, you need to check if it satisfies the equation of the circle. If the point's coordinates satisfy the equation, then it lies on the circle.

Q: What is the relationship between the radius and the distance from the center to a point on the circle?

A: The radius of a circle is the distance from the center to any point on the circle. This means that the radius is always equal to the distance from the center to any point on the circle.

Q: Can a point have a different distance from the center than the radius of the circle?

A: No, a point cannot have a different distance from the center than the radius of the circle. If a point has a different distance from the center, then it does not lie on the circle.

Q: What is the significance of the point Q(10, 0) in the context of the circle?

A: The point Q(10, 0) is a point that lies on the circle. It is used to find the radius of the circle and to determine which points also lie on the circle.

Q: How do you determine which point lies on the circle among multiple options?

A: To determine which point lies on the circle among multiple options, you need to check if each point satisfies the equation of the circle. If a point's coordinates satisfy the equation, then it lies on the circle.

Q: What is the final answer to the problem of which point lies on the circle?

A: The final answer is A(5, 5√3). This point lies on the circle, while the other points do not.

Key Takeaways

The equation of a circle centered at the origin is x^2 + y^2 = r^2.
The radius of a circle can be found using the distance formula.
To determine if a point lies on a circle, you need to check if it satisfies the equation of the circle.
The radius of a circle is the distance from the center to any point on the circle.

Final Answer

The final answer is A(5, 5√3).