Which Equation Represents A Circle With A Center At { (-3, -5)$}$ And A Radius Of 6 Units?A. { (x-3)^2 + (y-5)^2 = 6$}$ B. { (x-3)^2 + (y-5)^2 = 36$}$ C. { (x+3)^2 + (y+5)^2 = 6$}$ D. [$(x+3)^2 + (y+5)^2
Circles are fundamental geometric shapes that play a crucial role in mathematics, particularly in geometry and trigonometry. A circle is defined as the set of all points in a plane that are equidistant from a fixed point, known as the center. In this article, we will explore the equation of a circle and determine which equation represents a circle with a center at (-3, -5) and a radius of 6 units.
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, where (x, y) are the coordinates of any point on the circle.
Determining the Equation of a Circle
To determine the equation of a circle, we need to know the center and the radius. In this case, the center is given as (-3, -5) and the radius is 6 units.
Substituting the Values into the Equation
Substituting the values of the center and the radius into the general equation of a circle, we get:
(x - (-3))^2 + (y - (-5))^2 = 6^2
Simplifying the equation, we get:
(x + 3)^2 + (y + 5)^2 = 36
Comparing the Options
Now, let's compare the derived equation with the given options:
A. (x-3)^2 + (y-5)^2 = 6 B. (x-3)^2 + (y-5)^2 = 36 C. (x+3)^2 + (y+5)^2 = 6 D. (x+3)^2 + (y+5)^2 = 36
Conclusion
Based on the derived equation, the correct answer is:
D. (x+3)^2 + (y+5)^2 = 36
This equation represents a circle with a center at (-3, -5) and a radius of 6 units.
Why is this Equation Correct?
The equation (x+3)^2 + (y+5)^2 = 36 is correct because it satisfies the conditions of a circle:
- The center is (-3, -5), which is the same as the given center.
- The radius is 6 units, which is the same as the given radius.
- The equation is in the form (x - h)^2 + (y - k)^2 = r^2, which is the general equation of a circle.
Common Mistakes to Avoid
When determining the equation of a circle, it's essential to avoid common mistakes, such as:
- Not substituting the values of the center and the radius correctly.
- Not simplifying the equation correctly.
- Not comparing the derived equation with the given options correctly.
Real-World Applications
Circles have numerous real-world applications, including:
- Geometry and trigonometry: Circles are used to calculate distances, angles, and areas.
- Physics: Circles are used to describe the motion of objects, such as the orbit of planets.
- Engineering: Circles are used to design and build structures, such as bridges and buildings.
Conclusion
In this article, we will answer some frequently asked questions about the equation of a circle.
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 determine the equation of a circle?
A: To determine the equation of a circle, you need to know the center and the radius. You can use the general equation of a circle and substitute the values of the center and the radius.
Q: What is the significance of the center and the radius in the equation of a circle?
A: The center (h, k) represents the point in the plane that is equidistant from all points on the circle. The radius r represents the distance from the center to any point on the circle.
Q: How do I simplify the equation of a circle?
A: To simplify the equation of a circle, you can use the following steps:
- Substitute the values of the center and the radius into the general equation.
- Expand the squared terms.
- Combine like terms.
- Simplify the equation.
Q: What are some common mistakes to avoid when determining the equation of a circle?
A: Some common mistakes to avoid when determining the equation of a circle include:
- Not substituting the values of the center and the radius correctly.
- Not simplifying the equation correctly.
- Not comparing the derived equation with the given options correctly.
Q: What are some real-world applications of the equation of a circle?
A: The equation of a circle has numerous real-world applications, including:
- Geometry and trigonometry: Circles are used to calculate distances, angles, and areas.
- Physics: Circles are used to describe the motion of objects, such as the orbit of planets.
- Engineering: Circles are used to design and build structures, such as bridges and buildings.
Q: How do I graph a circle using its equation?
A: To graph a circle using its equation, you can use the following steps:
- Identify the center and the radius of the circle.
- Plot the center of the circle on a coordinate plane.
- Draw a circle with the center at the plotted point and the radius equal to the given radius.
Q: What are some common types of circles?
A: Some common types of circles include:
- A circle with a center at the origin (0, 0) and a radius of r is called a unit circle.
- A circle with a center at (h, k) and a radius of r is called a circle with center (h, k) and radius r.
- A circle with a center at (h, k) and a radius of r that is tangent to a line is called a circle with center (h, k) and radius r that is tangent to the line.
Q: How do I find the equation of a circle that is tangent to a line?
A: To find the equation of a circle that is tangent to a line, you can use the following steps:
- Identify the center and the radius of the circle.
- Identify the equation of the line that the circle is tangent to.
- Use the equation of the line and the center and radius of the circle to find the equation of the circle.
Conclusion
In conclusion, the equation of a circle is a fundamental concept in mathematics that has numerous real-world applications. By understanding the general equation of a circle and how to determine its equation, you can solve a wide range of problems in geometry, trigonometry, physics, and engineering.