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 =
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 represented 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 with a center at (-4, 9) and a diameter of 10 units.
The Center of the Circle
The center of the circle is given as (-4, 9). This means that the x-coordinate of the center is -4 and the y-coordinate is 9.
The Diameter of the Circle
The diameter of the circle is given as 10 units. The diameter is twice the radius, so the radius of the circle is 10/2 = 5 units.
The Equation of a Circle
The equation of a circle can be represented 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 case, the center of the circle is (-4, 9) and the radius is 5 units.
Substituting the Values
Substituting the values of the center and the radius into the equation of a circle, we get:
(x - (-4))^2 + (y - 9)^2 = 5^2
Simplifying the equation, we get:
(x + 4)^2 + (y - 9)^2 = 25
Comparing the Options
Now, let's compare the equation we derived with the options given:
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 we derived matches option B.
Conclusion
In conclusion, the equation that represents a 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.
Why is this Equation Correct?
This equation is correct because it represents a circle with a center at (-4, 9) and a radius of 5 units. The radius is half the diameter, so the radius is 10/2 = 5 units. The equation (x + 4)^2 + (y - 9)^2 = 25 represents a circle with a center at (-4, 9) and a radius of 5 units.
What is the Diameter of the Circle?
The diameter of the circle is 10 units. This is twice the radius, so the radius is 10/2 = 5 units.
What is the Radius of the Circle?
The radius of the circle is 5 units. This is half the diameter, so the diameter is 10 units.
What is the Center of the Circle?
The center of the circle is (-4, 9). This is the point that is equidistant from all points on the circle.
What is the Equation of a Circle?
The equation of a circle can be represented 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.
How to Find the Equation of a Circle
To find the equation of a circle, you need to know the center and the radius of the circle. The center is the point that is equidistant from all points on the circle, and the radius is the distance from the center to any point on the circle.
What is the Formula for the Equation of a Circle?
The formula for the equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
How to Use the Formula
To use the formula, you need to substitute the values of the center and the radius into the equation. The center is the point that is equidistant from all points on the circle, and the radius is the distance from the center to any point on the circle.
What are the Steps to Find the Equation of a Circle?
The steps to find the equation of a circle are:
- Find the center of the circle.
- Find the radius of the circle.
- Substitute the values of the center and the radius into the equation of a circle.
What is the Importance of the Equation of a Circle?
The equation of a circle is important because it represents a set of points that are all equidistant from a central point. This is useful in many areas of mathematics and science, such as geometry, trigonometry, and physics.
What are the Applications of the Equation of a Circle?
The equation of a circle has many applications in mathematics and science, such as:
- Geometry: The equation of a circle is used to find the area and circumference of a circle.
- Trigonometry: The equation of a circle is used to find the sine, cosine, and tangent of an angle.
- Physics: The equation of a circle is used to describe the motion of an object in a circular path.
Conclusion
In conclusion, the equation that represents a circle with a center at (-4, 9) and a diameter of 10 units is:
(x + 4)^2 + (y - 9)^2 = 25
In our previous article, we explored the equation of a circle and how to find the equation of a circle. In this article, we will answer some frequently asked questions about the equation of a circle.
Q: What is the equation of a circle?
A: The equation of a circle is a mathematical equation that represents a set of points that are all equidistant from a central point. The equation of a circle can be represented 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.
Q: How do I find the equation of a circle?
A: To find the equation of a circle, you need to know the center and the radius of the circle. The center is the point that is equidistant from all points on the circle, and the radius is the distance from the center to any point on the circle. You can find the equation of a circle by substituting the values of the center and the radius into the equation of a circle.
Q: What is the center of a circle?
A: The center of a circle is the point that is equidistant from all points on the circle. It is the point that is at the center of the circle.
Q: What is the radius of a circle?
A: The radius of a circle is the distance from the center to any point on the circle. It is the length of the line segment that connects the center to any point on the circle.
Q: How do I find the center and radius of a circle?
A: To find the center and radius of a circle, you need to know the coordinates of the center and the length of the radius. You can find the center and radius of a circle by using the equation of a circle and substituting the values of the coordinates and the length of the radius.
Q: What is the formula for the equation of a circle?
A: The formula for the equation of a circle is (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 use the formula for the equation of a circle?
A: To use the formula for the equation of a circle, you need to substitute the values of the center and the radius into the equation. The center is the point that is equidistant from all points on the circle, and the radius is the distance from the center to any point on the circle.
Q: What are the steps to find the equation of a circle?
A: The steps to find the equation of a circle are:
- Find the center of the circle.
- Find the radius of the circle.
- Substitute the values of the center and the radius into the equation of a circle.
Q: What is the importance of the equation of a circle?
A: The equation of a circle is important because it represents a set of points that are all equidistant from a central point. This is useful in many areas of mathematics and science, such as geometry, trigonometry, and physics.
Q: What are the applications of the equation of a circle?
A: The equation of a circle has many applications in mathematics and science, such as:
- Geometry: The equation of a circle is used to find the area and circumference of a circle.
- Trigonometry: The equation of a circle is used to find the sine, cosine, and tangent of an angle.
- Physics: The equation of a circle is used to describe the motion of an object in a circular path.
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 the formula A = πr^2, and the circumference of a circle is given by the formula C = 2πr.
Q: Can I use the equation of a circle to find the sine, cosine, and tangent of an angle?
A: Yes, you can use the equation of a circle to find the sine, cosine, and tangent of an angle. The sine, cosine, and tangent of an angle can be found using the equation of a circle and the coordinates of the center and the radius.
Q: Can I use the equation of a circle to describe the motion of an object in a circular path?
A: Yes, you can use the equation of a circle to describe the motion of an object in a circular path. The equation of a circle can be used to describe the motion of an object in a circular path by using the equation of a circle and the coordinates of the center and the radius.
Conclusion
In conclusion, the equation of a circle is a mathematical equation that represents a set of points that are all equidistant from a central point. The equation of a circle can be represented 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. We hope that this article has helped you to understand the equation of a circle and how to use it to find the area and circumference of a circle, the sine, cosine, and tangent of an angle, and the motion of an object in a circular path.