Which Equation Represents A Circle Centered At \[$(3,5)\$\] And Passing Through The Point \[$(-2,9)\$\]?A. \[$(x+3)^2 + (y+5)^2 = 17\$\]B. \[$(x+3)^2 + (y+5)^2 = 41\$\]C. \[$(x-3)^2 + (y-5)^2 = 41\$\]D.

Feb 28, 2025 by ADMIN 203 views

Introduction

In mathematics, a circle is a set of points that are equidistant from a central point called the center. The equation of a circle can be written in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. In this article, we will explore which equation represents a circle centered at (3, 5) and passing through the point (-2, 9).

Understanding the Problem

To solve this problem, we need to understand the properties of a circle and how to write its equation. The center of the circle is given as (3, 5), and the point through which the circle passes is given as (-2, 9). We need to find the equation of the circle that satisfies these conditions.

Step 1: Find the Distance Between the Center and the Point

The distance between the center (3, 5) and the point (-2, 9) can be found using the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) is the center (3, 5) and (x2, y2) is the point (-2, 9).

d = √((-2 - 3)^2 + (9 - 5)^2) d = √((-5)^2 + (4)^2) d = √(25 + 16) d = √41

Step 2: Write the Equation of the Circle

Now that we have the distance between the center and the point, we can write the equation of the circle. The equation of a circle with center (h, k) and radius r is given by:

(x - h)^2 + (y - k)^2 = r^2

In this case, the center is (3, 5) and the radius is √41. Therefore, the equation of the circle is:

(x - 3)^2 + (y - 5)^2 = (√41)^2

Simplifying the equation, we get:

(x - 3)^2 + (y - 5)^2 = 41

Conclusion

Based on the calculations above, the equation that represents a circle centered at (3, 5) and passing through the point (-2, 9) is:

(x - 3)^2 + (y - 5)^2 = 41

This equation satisfies the conditions given in the problem, and it represents a circle with center (3, 5) and radius √41.

Answer

The correct answer is:

C. (x - 3)^2 + (y - 5)^2 = 41

Discussion

This problem requires a good understanding of the properties of a circle and how to write its equation. The distance formula is used to find the distance between the center and the point, and then the equation of the circle is written using the distance as the radius. This problem is a good example of how to apply mathematical concepts to real-world problems.

References

[1] "Equation of a Circle" by Math Open Reference
[2] "Distance Formula" by Math Is Fun
[3] "Properties of a Circle" by Khan Academy
Frequently Asked Questions (FAQs) About Circles and Their Equations ====================================================================

Q: What is the general equation of a circle?

A: The general equation of a circle is given by (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

Q: How do I find the center and radius of a circle from its equation?

A: To find the center and radius of a circle from its equation, you need to rewrite the equation in the standard form (x - h)^2 + (y - k)^2 = r^2. The values of h and k will give you the coordinates of the center, and the value of r will give you the radius.

Q: What is the distance between the center and a point on the circle?

A: The distance between the center and a point on the circle is equal to the radius of the circle.

Q: How do I find the equation of a circle that passes through two points?

A: To find the equation of a circle that passes through two points, you need to find the center of the circle and the radius. You can use the distance formula to find the distance between the two points, and then use the equation of a circle to find the center and radius.

Q: What is the relationship between the equation of a circle and its graph?

A: The equation of a circle represents the set of all points that are equidistant from a central point called the center. The graph of a circle is a set of points that satisfy the equation of the circle.

Q: Can a circle have a negative radius?

A: No, a circle cannot have a negative radius. The radius of a circle is always a positive value.

Q: Can a circle have a zero radius?

A: No, a circle cannot have a zero radius. A circle with a zero radius would be a single point, not a circle.

Q: How do I graph a circle?

A: To graph a circle, you need to plot the center of the circle and then draw a circle with the given radius.

Q: What are some real-world applications of circles and their equations?

A: Circles and their equations have many real-world applications, such as:

Designing circular shapes for architecture and engineering
Calculating distances and angles in navigation and surveying
Modeling the motion of objects in physics and engineering
Creating art and designs that involve circular shapes

Q: Can I use a calculator to find the equation of a circle?

A: Yes, you can use a calculator to find the equation of a circle. Many calculators have built-in functions for finding the equation of a circle given the center and radius.

Q: How do I use the equation of a circle to solve problems?

A: To use the equation of a circle to solve problems, you need to understand the properties of a circle and how to apply the equation to real-world situations. You can use the equation to find the distance between the center and a point on the circle, or to find the area and circumference of the circle.

Q: Can I use the equation of a circle to find the area and circumference of a circle?

A: Yes, you can use the equation of a circle to find the area and circumference of a circle. The area of a circle is given by A = πr^2, and the circumference is given by C = 2πr.