The Circle Below Is Centered At The Point { (3, -4)$}$ And Has A Radius Of Length 3. What Is Its Equation?A. { (x-3)^2 + (y+4)^2 = 9$}$B. { (x+4)^2 + (y-3)^2 = 3^2$}$C. { (x-3)^2 + (y-4)^2 = 9$} D . \[ D. \[ D . \[ (x-4)^2
Introduction
In mathematics, a circle is a set of points that are equidistant from a central point called the center. The distance from the center to any point on the circle is called the radius. In this article, we will explore the equation of a circle and how to derive it using the given information.
Understanding the Circle Equation
The general equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Given Information
We are given that the circle is centered at the point (3, -4) and has a radius of length 3. We need to find the equation of this circle.
Deriving the Equation
To derive the equation of the circle, we will substitute the given values into the general equation.
- The center of the circle is (3, -4), so h = 3 and k = -4.
- The radius of the circle is 3, so r = 3.
Substituting these values into the general equation, we get:
(x - 3)^2 + (y - (-4))^2 = 3^2
Simplifying the equation, we get:
(x - 3)^2 + (y + 4)^2 = 9
Comparing with the Options
Now, let's compare the derived equation with the given options:
- A. (x - 3)^2 + (y + 4)^2 = 9
- B. (x + 4)^2 + (y - 3)^2 = 3^2
- C. (x - 3)^2 + (y - 4)^2 = 9
- D. (x - 4)^2 + (y + 3)^2 = 3^2
The derived equation matches option A.
Conclusion
In this article, we derived the equation of a circle using the given information. We substituted the center and radius values into the general equation and simplified it to get the final equation. We compared the derived equation with the given options and found that it matches option A.
Key Takeaways
- The general equation of a circle is (x - h)^2 + (y - k)^2 = r^2.
- To derive the equation of a circle, substitute the center and radius values into the general equation.
- Simplify the equation to get the final equation.
Practice Problems
- A circle has a center at (2, 5) and a radius of length 4. Find its equation.
- A circle has a center at (-3, 2) and a radius of length 5. Find its equation.
- A circle has a center at (1, -1) and a radius of length 2. Find its equation.
Solutions
- (x - 2)^2 + (y - 5)^2 = 4^2
- (x + 3)^2 + (y - 2)^2 = 5^2
- (x - 1)^2 + (y + 1)^2 = 2^2
Circle Equation Q&A =====================
Frequently Asked Questions
Q: What is the general equation of a circle?
A: The general 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 derive the equation of a circle?
A: To derive the equation of a circle, substitute the center and radius values into the general equation and simplify it to get the final equation.
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 denoted as (h, k) in the general equation.
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 denoted as r in the general equation.
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, substitute the center and radius values into the general equation and simplify it to get the final equation.
Q: What is the significance of the squared terms in the circle equation?
A: The squared terms in the circle equation represent the squared distances from the center to any point on the circle. This is because the distance from the center to any point on the circle is always positive, and squaring it ensures that the equation is always true.
Q: Can I use the circle equation to find the center and radius of a circle?
A: Yes, you can use the circle equation to find the center and radius of a circle. By rearranging the equation, you can solve for the center and radius values.
Q: What is the relationship between the circle equation and the distance formula?
A: The circle equation is related to the distance formula, which is used to find the distance between two points. The circle equation can be derived from the distance formula by setting the distance equal to the radius.
Q: Can I use the circle equation to find the area of a circle?
A: Yes, you can use the circle equation to find the area of a circle. By rearranging the equation, you can solve for the radius value, and then use the formula for the area of a circle (A = πr^2) to find the area.
Q: What is the significance of the π term in the circle equation?
A: The π term in the circle equation represents the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.
Q: Can I use the circle equation to find the circumference of a circle?
A: Yes, you can use the circle equation to find the circumference of a circle. By rearranging the equation, you can solve for the radius value, and then use the formula for the circumference of a circle (C = 2πr) to find the circumference.
Q: What is the relationship between the circle equation and the trigonometric functions?
A: The circle equation is related to the trigonometric functions, which are used to describe the relationships between the angles and sides of triangles. The circle equation can be used to derive the trigonometric identities and formulas.
Q: Can I use the circle equation to solve problems involving circles?
A: Yes, you can use the circle equation to solve problems involving circles. By applying the equation and using algebraic manipulations, you can solve for the unknown values and find the solutions to the problems.
Conclusion
In this article, we have answered some of the most frequently asked questions about the circle equation. We have covered topics such as the general equation of a circle, deriving the equation, finding the center and radius, and using the equation to solve problems involving circles. We hope that this article has been helpful in clarifying any doubts you may have had about the circle equation.