Write An Equation In General Form Of The Circle With The Given Properties.Center At { (-2, 9)$}$ And Passing Through The Origin.
Introduction
In mathematics, a circle is a set of points that are equidistant from a central point called the center. The equation of a circle can be written in various forms, including standard form, general form, and parametric form. In this article, we will focus on writing the equation of a circle in general form, given its center and a point it passes through.
General Form of a Circle
The general form of a circle is given by the equation:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is the radius.
Given Properties
We are given that the center of the circle is at (-2, 9) and the circle passes through the origin (0, 0). We need to write the equation of the circle in general form using these properties.
Step 1: Substitute the Center into the General Form
Substituting the center (-2, 9) into the general form of the circle, we get:
(x - (-2))^2 + (y - 9)^2 = r^2
Simplifying the equation, we get:
(x + 2)^2 + (y - 9)^2 = r^2
Step 2: Use the Point (0, 0) to Find the Radius
Since the circle passes through the origin (0, 0), we can substitute x = 0 and y = 0 into the equation to find the radius.
(0 + 2)^2 + (0 - 9)^2 = r^2
Simplifying the equation, we get:
4 + 81 = r^2
r^2 = 85
Step 3: Write the Equation in General Form
Now that we have found the radius, we can write the equation of the circle in general form:
(x + 2)^2 + (y - 9)^2 = 85
Conclusion
In this article, we have written the equation of a circle in general form, given its center and a point it passes through. We have used the properties of the circle to substitute the center and the point into the general form, and then simplified the equation to find the radius. The final equation of the circle in general form is:
(x + 2)^2 + (y - 9)^2 = 85
Properties of the Circle
- Center: The center of the circle is at (-2, 9).
- Radius: The radius of the circle is √85.
- Equation: The equation of the circle in general form is (x + 2)^2 + (y - 9)^2 = 85.
Real-World Applications
The equation of a circle has many real-world applications, including:
- Geometry: The equation of a circle is used to describe the shape and size of a circle.
- Trigonometry: The equation of a circle is used to solve problems involving right triangles and trigonometric functions.
- Physics: The equation of a circle is used to describe the motion of objects in circular paths.
Tips and Tricks
- Use the Center: When writing the equation of a circle, use the center to substitute into the general form.
- Use the Point: When writing the equation of a circle, use a point it passes through to find the radius.
- Simplify the Equation: Simplify the equation to find the radius and write the equation in general form.
Practice Problems
- Write the equation of a circle in general form, given its center at (3, 4) and passing through the point (0, 0).
- Write the equation of a circle in general form, given its center at (-5, 2) and passing through the point (0, 0).
- Write the equation of a circle in general form, given its center at (1, 6) and passing through the point (0, 0).
Conclusion
Q: What is the general form of a circle?
A: The general form of a circle is given by the equation:
(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 write the equation of a circle in general form?
A: To write the equation of a circle in general form, you need to know the center of the circle and a point it passes through. You can then substitute the center and the point into the general form and simplify the equation to find the radius.
Q: What is the center of the circle in the given problem?
A: The center of the circle is at (-2, 9).
Q: What is the radius of the circle in the given problem?
A: The radius of the circle is √85.
Q: How do I find the radius of the circle?
A: To find the radius of the circle, you can substitute a point it passes through into the equation and solve for the radius.
Q: What is the equation of the circle in general form?
A: The equation of the circle in general form is:
(x + 2)^2 + (y - 9)^2 = 85
Q: What are some real-world applications of the equation of a circle?
A: The equation of a circle has many real-world applications, including:
- Geometry: The equation of a circle is used to describe the shape and size of a circle.
- Trigonometry: The equation of a circle is used to solve problems involving right triangles and trigonometric functions.
- Physics: The equation of a circle is used to describe the motion of objects in circular paths.
Q: What are some tips and tricks for writing the equation of a circle?
A: Here are some tips and tricks for writing the equation of a circle:
- Use the Center: When writing the equation of a circle, use the center to substitute into the general form.
- Use the Point: When writing the equation of a circle, use a point it passes through to find the radius.
- Simplify the Equation: Simplify the equation to find the radius and write the equation in general form.
Q: What are some practice problems for writing the equation of a circle?
A: Here are some practice problems for writing the equation of a circle:
- Write the equation of a circle in general form, given its center at (3, 4) and passing through the point (0, 0).
- Write the equation of a circle in general form, given its center at (-5, 2) and passing through the point (0, 0).
- Write the equation of a circle in general form, given its center at (1, 6) and passing through the point (0, 0).
Q: How do I know if the equation of a circle is in general form?
A: The equation of a circle is in general form if it is written 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: What are some common mistakes to avoid when writing the equation of a circle?
A: Here are some common mistakes to avoid when writing the equation of a circle:
- Not using the center: Make sure to use the center of the circle to substitute into the general form.
- Not using a point it passes through: Make sure to use a point the circle passes through to find the radius.
- Not simplifying the equation: Make sure to simplify the equation to find the radius and write the equation in general form.
Conclusion
In this article, we have answered some frequently asked questions about the equation of a circle in general form. We have discussed the general form of a circle, how to write the equation of a circle in general form, and some real-world applications of the equation of a circle. We have also provided some tips and tricks for writing the equation of a circle and some practice problems to help you practice.