Find An Equation Of The Circle That Has Its Center At \[$(1, 6)\$\] And Passes Through \[$(5, -3)\$\].

Mar 1, 2025 by ADMIN 103 views

Introduction

In mathematics, a circle is a set of points that are all equidistant from a central point called the center. The equation of a circle can be used to describe the circle's position and size on a coordinate plane. In this article, we will discuss how to find the equation of a circle that has its center at (1, 6) and passes through (5, -3).

What is the Equation of a Circle?

The equation of a circle is given by the formula:

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

where (h, k) is the center of the circle and r is the radius.

Finding the Equation of the Circle

To find the equation of the circle, we need to find the radius. We can do this by using the distance formula to find the distance between the center (1, 6) and the point (5, -3).

Distance Formula

The distance formula is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) is the center of the circle and (x2, y2) is the point on the circle.

Calculating the Distance

We can plug in the values into the distance formula to get:

d = √((5 - 1)^2 + (-3 - 6)^2) d = √((4)^2 + (-9)^2) d = √(16 + 81) d = √97

Finding the Radius

The radius is the distance between the center and the point on the circle. Therefore, the radius is:

r = √97

Finding the Equation of the Circle

Now that we have the radius, we can plug it into the equation of the circle formula:

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

We know that the center is (1, 6) and the radius is √97. Therefore, the equation of the circle is:

(x - 1)^2 + (y - 6)^2 = (√97)^2

Simplifying the equation, we get:

(x - 1)^2 + (y - 6)^2 = 97

Conclusion

In this article, we discussed how to find the equation of a circle that has its center at (1, 6) and passes through (5, -3). We used the distance formula to find the radius and then plugged it into the equation of the circle formula to get the final equation. The equation of the circle is:

(x - 1)^2 + (y - 6)^2 = 97

This equation describes the circle's position and size on the coordinate plane.

Example Use Cases

Finding the equation of a circle that passes through three given points.
Determining the radius of a circle given its center and a point on the circle.
Graphing a circle on a coordinate plane using its equation.

Step-by-Step Solution

Find the distance between the center and the point on the circle using the distance formula.
Use the distance as the radius in the equation of the circle formula.
Simplify the equation to get the final equation of the circle.

Common Mistakes

Forgetting to square the radius in the equation of the circle formula.
Not using the correct values for the center and the point on the circle.
Not simplifying the equation to get the final equation of the circle.

Tips and Tricks

Use the distance formula to find the radius of the circle.
Plug the radius into the equation of the circle formula.
Simplify the equation to get the final equation of the circle.

Real-World Applications

Finding the equation of a circle that represents a satellite's orbit around the Earth.
Determining the radius of a circle that represents a city's boundary.
Graphing a circle on a coordinate plane to represent a shape in a design.

Conclusion

In conclusion, finding the equation of a circle that has its center at (1, 6) and passes through (5, -3) requires using the distance formula to find the radius and then plugging it into the equation of the circle formula. The final equation of the circle is:

(x - 1)^2 + (y - 6)^2 = 97

This equation describes the circle's position and size on the coordinate plane.

Introduction

In our previous article, we discussed how to find the equation of a circle that has its center at (1, 6) and passes through (5, -3). In this article, we will answer some frequently asked questions related to finding the equation of a circle.

Q&A

Q1: What is the equation of a circle?

A1: The equation of a circle is given by the formula:

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

where (h, k) is the center of the circle and r is the radius.

Q2: How do I find the radius of a circle?

A2: To find the radius of a circle, you can use the distance formula to find the distance between the center and a point on the circle.

Q3: What is the distance formula?

A3: The distance formula is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) is the center of the circle and (x2, y2) is the point on the circle.

Q4: How do I find the equation of a circle that passes through three given points?

A4: To find the equation of a circle that passes through three given points, you can use the following steps:

Find the distance between each pair of points.
Use the distances to find the radius of the circle.
Plug the radius into the equation of the circle formula.

Q5: What is the significance of the center of a circle?

A5: The center of a circle is the point that is equidistant from all points on the circle. It is also the point that is used to find the equation of the circle.

Q6: How do I determine the radius of a circle given its center and a point on the circle?

A6: To determine the radius of a circle given its center and a point on the circle, you can use the distance formula to find the distance between the center and the point.

Q7: What is the equation of a circle that has its center at (1, 6) and passes through (5, -3)?

A7: The equation of a circle that has its center at (1, 6) and passes through (5, -3) is:

(x - 1)^2 + (y - 6)^2 = 97

Q8: How do I graph a circle on a coordinate plane?

A8: To graph a circle on a coordinate plane, you can use the equation of the circle to find the x and y coordinates of the points on the circle.

Q9: What are some real-world applications of finding the equation of a circle?

A9: Some real-world applications of finding the equation of a circle include:

Finding the equation of a circle that represents a satellite's orbit around the Earth.
Determining the radius of a circle that represents a city's boundary.
Graphing a circle on a coordinate plane to represent a shape in a design.

Q10: What are some common mistakes to avoid when finding the equation of a circle?

A10: Some common mistakes to avoid when finding the equation of a circle include:

Forgetting to square the radius in the equation of the circle formula.
Not using the correct values for the center and the point on the circle.
Not simplifying the equation to get the final equation of the circle.

Conclusion

(x - 1)^2 + (y - 6)^2 = 97

This equation describes the circle's position and size on the coordinate plane.