Which Equation Represents A Circle With A Center At { (-4,9)$}$ And A Diameter Of 10 Units?A. { (x-9) 2+(y+4) 2=25$}$B. { (x+4) 2+(y-9) 2=25$}$C. { (x-9) 2+(y+4) 2=100$}$D. { (x+4) 2+(y-9) 2=100$}$
The equation of a circle is a fundamental concept in mathematics, and it is essential to understand how to represent a circle using an equation. In this article, we will explore the equation of a circle and how to determine which equation represents a circle with a specific center and diameter.
The Standard Equation of a Circle
The standard 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.
Given Information
We are given that the center of the circle is at (-4, 9) and the diameter is 10 units. Since the diameter is twice the radius, the radius is 5 units.
Determining the Equation
To determine which equation represents the circle, we need to substitute the given values into the standard equation of a circle.
Step 1: Substitute the Center Coordinates
The center of the circle is at (-4, 9), so we substitute h = -4 and k = 9 into the standard equation:
(x - (-4))^2 + (y - 9)^2 = r^2
Simplifying the equation, we get:
(x + 4)^2 + (y - 9)^2 = r^2
Step 2: Substitute the Radius
The radius of the circle is 5 units, so we substitute r = 5 into the equation:
(x + 4)^2 + (y - 9)^2 = 5^2
Simplifying the equation, we get:
(x + 4)^2 + (y - 9)^2 = 25
Comparing the Equations
Now that we have the equation of the circle, we can compare it with the given options:
A. (x - 9)^2 + (y + 4)^2 = 25 B. (x + 4)^2 + (y - 9)^2 = 25 C. (x - 9)^2 + (y + 4)^2 = 100 D. (x + 4)^2 + (y - 9)^2 = 100
Conclusion
Based on our calculations, the correct equation that represents the circle with a center at (-4, 9) and a diameter of 10 units is:
(x + 4)^2 + (y - 9)^2 = 25
This equation matches option B.
Final Answer
In the previous article, we explored the equation of a circle and how to determine which equation represents a circle with a specific center and diameter. In this article, we will answer some frequently asked questions about the equation of a circle.
Q: What is the standard equation of a circle?
A: The standard 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 determine the equation of a circle if I know the center and diameter?
A: To determine the equation of a circle, you need to substitute the given values into the standard equation of a circle. First, substitute the center coordinates (h, k) into the equation. Then, substitute the radius (r) into the equation. Finally, simplify the equation to get the final answer.
Q: What is the difference between the center and the radius of a circle?
A: The center of a circle is the point at the center of the circle, and it is denoted by (h, k). The radius of a circle is the distance from the center to any point on the circle, and it is denoted by r.
Q: How do I find the radius of a circle if I know the diameter?
A: The radius of a circle is half the diameter. If you know the diameter, you can find the radius by dividing the diameter by 2.
Q: What is the equation of a circle with a center at (0, 0) and a radius of 3?
A: To find the equation of a circle with a center at (0, 0) and a radius of 3, substitute the center coordinates (h, k) = (0, 0) and the radius (r) = 3 into the standard equation of a circle:
(x - 0)^2 + (y - 0)^2 = 3^2
Simplifying the equation, we get:
x^2 + y^2 = 9
Q: What is the equation of a circle with a center at (2, 3) and a radius of 4?
A: To find the equation of a circle with a center at (2, 3) and a radius of 4, substitute the center coordinates (h, k) = (2, 3) and the radius (r) = 4 into the standard equation of a circle:
(x - 2)^2 + (y - 3)^2 = 4^2
Simplifying the equation, we get:
(x - 2)^2 + (y - 3)^2 = 16
Q: How do I graph a circle using its equation?
A: To graph a circle using its equation, you need to plot the center of the circle and then draw a circle with the given radius. You can use a compass to draw the circle.
Q: What is the equation of a circle with a center at (-2, 1) and a diameter of 6?
A: To find the equation of a circle with a center at (-2, 1) and a diameter of 6, first find the radius by dividing the diameter by 2:
r = 6/2 = 3
Then, substitute the center coordinates (h, k) = (-2, 1) and the radius (r) = 3 into the standard equation of a circle:
(x - (-2))^2 + (y - 1)^2 = 3^2
Simplifying the equation, we get:
(x + 2)^2 + (y - 1)^2 = 9
Conclusion
In this article, we answered some frequently asked questions about the equation of a circle. We hope that this article has helped you to understand the equation of a circle and how to use it to solve problems.