This Circle Is Centered At The Origin, And The Length Of Its Radius Is 8. What Is The Circle's Equation?A. X 8 + Y 8 = 64 X^8 + Y^8 = 64 X 8 + Y 8 = 64 B. X + Y = 8 X + Y = 8 X + Y = 8 C. X 2 + Y 2 = 8 X^2 + Y^2 = 8 X 2 + Y 2 = 8 D. X 2 + Y 2 = 64 X^2 + Y^2 = 64 X 2 + Y 2 = 64
In mathematics, a circle is a set of points that are equidistant from a central point known as the center. The distance from the center to any point on the circle is called the radius. When the circle is centered at the origin, its equation can be easily determined using the distance formula. In this article, we will explore the equation of a circle centered at the origin with a radius of 8.
The General Equation of a Circle
The general equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
This equation represents a circle with center (h, k) and radius r. When the circle is centered at the origin (0, 0), the equation simplifies to:
x^2 + y^2 = r^2
The Equation of a Circle Centered at the Origin
Given that the circle is centered at the origin and the length of its radius is 8, we can substitute these values into the simplified equation:
x^2 + y^2 = r^2 x^2 + y^2 = 8^2 x^2 + y^2 = 64
Therefore, the equation of the circle is x^2 + y^2 = 64.
Analyzing the Answer Choices
Now that we have determined the equation of the circle, let's analyze the answer choices:
A. This option is incorrect because the equation of a circle is not a sum of powers of x and y, but rather a sum of squares.
B. This option is incorrect because the equation of a circle is not a linear equation, but rather a quadratic equation.
C. This option is incorrect because the radius of the circle is 8, not 2√2.
D. This option is correct because it matches the equation we derived earlier.
Conclusion
In conclusion, the equation of a circle centered at the origin with a radius of 8 is x^2 + y^2 = 64. This equation represents a circle with center (0, 0) and radius 8. We analyzed the answer choices and determined that option D is the correct answer.
Key Takeaways
- The general equation of a circle is (x - h)^2 + (y - k)^2 = r^2.
- When the circle is centered at the origin, the equation simplifies to x^2 + y^2 = r^2.
- The equation of a circle centered at the origin with a radius of 8 is x^2 + y^2 = 64.
Frequently Asked Questions
Q: What is the equation of a circle centered at the origin with a radius of 8? A: The equation of the circle is x^2 + y^2 = 64.
Q: How do I determine the equation of a circle? A: To determine the equation of a circle, you need to know the center and radius of the circle. If the circle is centered at the origin, the equation simplifies to x^2 + y^2 = r^2.
In this article, we will continue to explore the equation of a circle centered at the origin with a radius of 8. We will also answer some frequently asked questions about the equation of a circle.
Q: What is the equation of a circle centered at the origin with a radius of 8?
A: The equation of the circle is x^2 + y^2 = 64.
Q: How do I determine the equation of a circle?
A: To determine the equation of a circle, you need to know the center and radius of the circle. If the circle is centered at the origin, the equation simplifies to x^2 + y^2 = r^2.
Q: What is the difference between the general equation of a circle and the equation of a circle centered at the origin?
A: The general equation of a circle is (x - h)^2 + (y - k)^2 = r^2, while the equation of a circle centered at the origin is x^2 + y^2 = r^2.
Q: Can I use the equation of a circle to find the center and radius of a circle?
A: Yes, you can use the equation of a circle to find the center and radius of a circle. If the equation is in the form x^2 + y^2 = r^2, the center is at the origin (0, 0) and the radius is √r^2.
Q: How do I graph a circle using its equation?
A: To graph a circle using its equation, you need to know the center and radius of the circle. If the equation is in the form x^2 + y^2 = r^2, you can graph the circle by plotting the center and drawing a circle with the given radius.
Q: Can I use the equation of a circle to find the area of a circle?
A: Yes, you can use the equation of a circle to find the area of a circle. The area of a circle is given by the formula A = πr^2, where r is the radius of the circle.
Q: How do I find the equation of a circle given its center and radius?
A: To find the equation of a circle given its center and radius, you can use the general equation of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
Q: Can I use the equation of a circle to find the distance between two points on the circle?
A: Yes, you can use the equation of a circle to find the distance between two points on the circle. If the two points are (x1, y1) and (x2, y2), you can use the distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2).
Q: How do I find the equation of a circle given its diameter?
A: To find the equation of a circle given its diameter, you need to know the center of the circle. If the diameter is given, you can find the radius by dividing the diameter by 2. Then, you can use the general equation of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
Q: Can I use the equation of a circle to find the equation of a tangent line to the circle?
A: Yes, you can use the equation of a circle to find the equation of a tangent line to the circle. If the tangent line is perpendicular to the radius at the point of tangency, you can use the slope of the radius to find the slope of the tangent line. Then, you can use the point-slope form of a line to find the equation of the tangent line.
Conclusion
In conclusion, the equation of a circle is a powerful tool for finding the center and radius of a circle, graphing a circle, finding the area of a circle, and finding the distance between two points on the circle. We hope this article has been helpful in answering your questions about the equation of a circle.