Determine The Equation Of The Circle With Center { (9, -8)$}$ Containing The Point { (1, 7)$}$.
Determine the Equation of the Circle with Center (9, -8) Containing the Point (1, 7)
In mathematics, a circle is a set of points that are all equidistant from a central point known as the center. The equation of a circle can be determined using the center and a point on the circle. In this article, we will determine the equation of a circle with center (9, -8) containing the point (1, 7).
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.
Step 1: Identify the Center and the Point
The center of the circle is given as (9, -8) and the point on the circle is (1, 7).
Step 2: Calculate the Radius
To calculate the radius, we need to find the distance between the center and the point. We can use the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) is the center and (x2, y2) is the point.
Plugging in the values, we get:
d = √((1 - 9)^2 + (7 - (-8))^2) d = √((-8)^2 + (15)^2) d = √(64 + 225) d = √289 d = 17
So, the radius of the circle is 17.
Step 3: Write the Equation of the Circle
Now that we have the center and the radius, we can write the equation of the circle:
(x - 9)^2 + (y - (-8))^2 = 17^2
Simplifying the equation, we get:
(x - 9)^2 + (y + 8)^2 = 289
In this article, we determined the equation of a circle with center (9, -8) containing the point (1, 7). We used the distance formula to calculate the radius and then wrote the equation of the circle using the center and the radius.
Example Use Case
The equation of a circle can be used in various real-world applications such as:
- Calculating the area of a circle
- Finding the circumference of a circle
- Determining the distance between two points on a circle
- Calculating the volume of a sphere
Tips and Variations
- To find the equation of a circle with a given center and radius, use the formula: (x - h)^2 + (y - k)^2 = r^2
- To find the equation of a circle with a given point and center, use the distance formula to calculate the radius and then write the equation of the circle
- To find the equation of a circle with a given center and two points, use the distance formula to calculate the radius and then write the equation of the circle
Further Reading
For more information on circles and their equations, check out the following resources:
References
- Math Open Reference: Circle
- Purplemath: Circle
- Mathway: Circle
Determine the Equation of the Circle with Center (9, -8) Containing the Point (1, 7) - Q&A
In our previous article, we determined the equation of a circle with center (9, -8) containing the point (1, 7). In this article, we will answer some frequently asked questions related to the equation of a circle.
Q: What is the equation of a circle?
A: 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.
Q: How do I find the equation of a circle with a given center and radius?
A: To find the equation of a circle with a given center and radius, use the formula:
(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 equation of a circle with a given point and center?
A: To find the equation of a circle with a given point and center, use the distance formula to calculate the radius and then write the equation of the circle.
Q: What is the distance formula?
A: The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) is the center and (x2, y2) is the point.
Q: How do I calculate the radius of a circle?
A: To calculate the radius of a circle, use the distance formula to find the distance between the center and the point.
Q: What is the significance of the center and radius of a circle?
A: The center and radius of a circle are used to determine the equation of the circle.
Q: Can I find the equation of a circle with a given center and two points?
A: Yes, you can find the equation of a circle with a given center and two points by using the distance formula to calculate the radius and then writing the equation of the circle.
Q: What are some real-world applications of the equation of a circle?
A: Some real-world applications of the equation of a circle include:
- Calculating the area of a circle
- Finding the circumference of a circle
- Determining the distance between two points on a circle
- Calculating the volume of a sphere
Q: Where can I find more information on circles and their equations?
A: You can find more information on circles and their equations on the following websites:
In this article, we answered some frequently asked questions related to the equation of a circle. We hope that this article has provided you with a better understanding of the equation of a circle and its significance.
Example Use Case
The equation of a circle can be used in various real-world applications such as:
- Calculating the area of a circle
- Finding the circumference of a circle
- Determining the distance between two points on a circle
- Calculating the volume of a sphere
Tips and Variations
- To find the equation of a circle with a given center and radius, use the formula: (x - h)^2 + (y - k)^2 = r^2
- To find the equation of a circle with a given point and center, use the distance formula to calculate the radius and then write the equation of the circle
- To find the equation of a circle with a given center and two points, use the distance formula to calculate the radius and then write the equation of the circle
Further Reading
For more information on circles and their equations, check out the following resources: