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. \[$(x-3)^2 + (y+4)^2 = 9\$\]C. \[$(x-4)^2 + (y+3)^2 = 3^2\$\]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 from 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
This equation represents a circle with center (h, k) and radius r. The equation is derived from the fact that the distance between any point (x, y) on the circle and the center (h, k) is equal to the radius r.
Deriving the Circle Equation
To derive the circle equation, we need to use the distance formula, which is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, we want to find the distance between the point (x, y) and the center (h, k). We can plug in the values into the distance formula and simplify:
d = √((x - h)^2 + (y - k)^2)
Since the distance between any point on the circle and the center is equal to the radius r, we can set up the equation:
(x - h)^2 + (y - k)^2 = r^2
This is the general equation of a circle with center (h, k) and radius r.
Applying the Circle Equation
Now that we have the general equation of a circle, we can apply it to the given problem. The circle is centered at the point (3, -4) and has a radius of length 3. We can plug in these values into the equation:
(x - 3)^2 + (y - (-4))^2 = 3^2
Simplifying the equation, we get:
(x - 3)^2 + (y + 4)^2 = 9
This is the equation of the circle with center (3, -4) and radius 3.
Conclusion
In this article, we have explored the equation of a circle and how to derive it from the given information. We have also applied the circle equation to a specific problem and found the equation of the circle with center (3, -4) and radius 3. The equation of a circle is a fundamental concept in mathematics and has many practical applications in fields such as physics, engineering, and computer science.
The Final Answer
The final answer is:
(x - 3)^2 + (y - (-4))^2 = 3^2
Which simplifies to:
(x - 3)^2 + (y + 4)^2 = 9
This is the equation of the circle with center (3, -4) and radius 3.
Comparison of Options
Let's compare the options given in the problem:
A. (x - 3)^2 + (y - 4)^2 = 9 B. (x - 3)^2 + (y + 4)^2 = 9 C. (x - 4)^2 + (y + 3)^2 = 3^2 D. (x + 4)^2 + (y - 3)^2 = 3^2
Option A is incorrect because the center is (3, -4), not (3, 4).
Option B is correct because it matches the equation we derived.
Option C is incorrect because the center is (3, -4), not (4, 3).
Option D is incorrect because the center is (3, -4), not (-4, 3).
Therefore, the correct answer is:
B. (x - 3)^2 + (y + 4)^2 = 9
Q&A: Frequently Asked Questions
Q: What is the general equation of a circle? A: The general equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Q: How do I derive the circle equation? A: To derive the circle equation, you need to use the distance formula, which is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
You can plug in the values into the distance formula and simplify to get the circle equation.
Q: What is the center of the circle? A: The center of the circle is the point (h, k) in the equation (x - h)^2 + (y - k)^2 = r^2.
Q: What is the radius of the circle? A: The radius of the circle is the value r in the equation (x - h)^2 + (y - k)^2 = r^2.
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, you can plug in the values into the general equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
Q: What is the significance of the radius in the circle equation? A: The radius in the circle equation represents the distance from the center of the circle to any point on the circle.
Q: Can I have a circle with a negative radius? A: No, the radius of a circle cannot be negative. The radius is always a positive value.
Q: Can I have a circle with a zero radius? A: No, the radius of a circle cannot be zero. A circle with a zero radius would be a single point, not a circle.
Q: How do I graph a circle? A: To graph a circle, you can use the equation (x - h)^2 + (y - k)^2 = r^2 and plot the points (h, k) and (h ± r, k) and (h, k ± r).
Q: What is the relationship between the circle equation and the distance formula? A: The circle equation is derived from the distance formula. The distance formula is used to find the distance between two points, and the circle equation is used to find the equation of a circle.
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. The area of a circle is given by A = πr^2, where r is the radius of the circle.
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. The circumference of a circle is given by C = 2Ï€r, where r is the radius of the circle.
Conclusion
In this article, we have explored the equation of a circle and how to derive it from the given information. We have also answered some frequently asked questions about the circle equation. The circle equation is a fundamental concept in mathematics and has many practical applications in fields such as physics, engineering, and computer science.
The Final Answer
The final answer is:
(x - 3)^2 + (y + 4)^2 = 9
This is the equation of the circle with center (3, -4) and radius 3.
Comparison of Options
Let's compare the options given in the problem:
A. (x - 3)^2 + (y - 4)^2 = 9 B. (x - 3)^2 + (y + 4)^2 = 9 C. (x - 4)^2 + (y + 3)^2 = 3^2 D. (x + 4)^2 + (y - 3)^2 = 3^2
Option A is incorrect because the center is (3, -4), not (3, 4).
Option B is correct because it matches the equation we derived.
Option C is incorrect because the center is (3, -4), not (4, 3).
Option D is incorrect because the center is (3, -4), not (-4, 3).
Therefore, the correct answer is:
B. (x - 3)^2 + (y + 4)^2 = 9
This is the equation of the circle with center (3, -4) and radius 3.