Not identifying the center and radius of the circle

Oct 13, 2025 by magicalceylontours.com 60 views

What is the Radius of a Circle Whose Equation is $x^2 + y^2 - 10x + 6y + 18 = 0$ ?

Understanding the Equation of a Circle

The equation of a circle in standard form is given by $(x - h)^2 + (y - k)^2 = r^2$ , where $(h, k)$ is the center of the circle and $r$ is the radius. To find the radius of a circle from its equation, we need to rewrite the equation in standard form.

Rewriting the Equation in Standard Form

The given equation is $x^2 + y^2 - 10x + 6y + 18 = 0$ . To rewrite this equation in standard form, we need to complete the square for both the $x$ and $y$ terms.

x^2 - 10x + y^2 + 6y = -18

Completing the Square for $x$ and $y$ Terms

To complete the square for the $x$ term, we need to add $(10/2)^2 = 25$ to both sides of the equation. Similarly, to complete the square for the $y$ term, we need to add $(6/2)^2 = 9$ to both sides of the equation.

x^2 - 10x + 25 + y^2 + 6y + 9 = -18 + 25 + 9

Simplifying the Equation

Simplifying the equation, we get:

(x - 5)^2 + (y + 3)^2 = 16

Finding the Radius

Comparing this equation with the standard form of the equation of a circle, we can see that the radius is given by $r = \sqrt{16} = 4$ units.

Conclusion

Therefore, the radius of the circle whose equation is $x^2 + y^2 - 10x + 6y + 18 = 0$ is 4 units.

Key Takeaways

The equation of a circle in standard form is given by $(x - h)^2 + (y - k)^2 = r^2$ .
To find the radius of a circle from its equation, we need to rewrite the equation in standard form.
Completing the square for both the $x$ and $y$ terms is necessary to rewrite the equation in standard form.
The radius of a circle is given by $r = \sqrt{r^2}$ , where $r^2$ is the constant term on the right-hand side of the equation.

Frequently Asked Questions

What is the equation of a circle in standard form?
How do we find the radius of a circle from its equation?
What is the significance of completing the square for both the $x$ and $y$ terms?
How do we find the center of a circle from its equation?

Step-by-Step Solution

Rewrite the equation in standard form by completing the square for both the $x$ and $y$ terms.
Compare the rewritten equation with the standard form of the equation of a circle.
Identify the radius of the circle from the constant term on the right-hand side of the equation.

Real-World Applications

The equation of a circle is used to model real-world situations such as the path of a projectile, the shape of a mirror, and the design of a lens.
The radius of a circle is used to calculate the area and circumference of the circle.
The equation of a circle is used in computer graphics to create 2D and 3D shapes.

Mathematical Concepts

Completing the square
Standard form of the equation of a circle
Radius of a circle
Center of a circle

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Geometry" by Michael Artin

Additional Resources

Khan Academy: Completing the Square
MIT OpenCourseWare: Calculus
Wolfram Alpha: Equation of a Circle
Q&A: Understanding the Equation of a Circle

Frequently Asked Questions

What is the equation of a circle in standard form? The equation of a circle in standard form is given by $(x - h)^2 + (y - k)^2 = r^2$ , where $(h, k)$ is the center of the circle and $r$ is the radius.
How do we find the radius of a circle from its equation? To find the radius of a circle from its equation, we need to rewrite the equation in standard form by completing the square for both the $x$ and $y$ terms. The radius is then given by $r = \sqrt{r^2}$ , where $r^2$ is the constant term on the right-hand side of the equation.
What is the significance of completing the square for both the $x$ and $y$ terms? Completing the square for both the $x$ and $y$ terms is necessary to rewrite the equation in standard form. This allows us to identify the center and radius of the circle.
How do we find the center of a circle from its equation? To find the center of a circle from its equation, we need to rewrite the equation in standard form by completing the square for both the $x$ and $y$ terms. The center is then given by $(h, k)$ , where $h$ and $k$ are the values that are subtracted from $x$ and $y$ respectively.
What is the relationship between the equation of a circle and its graph? The equation of a circle represents the set of all points that are equidistant from a fixed point, called the center. The graph of the equation is a circle with the center at $(h, k)$ and the radius $r$ .
How do we use the equation of a circle to solve problems? We can use the equation of a circle to solve problems by substituting the values of the variables into the equation and solving for the unknowns. We can also use the equation to find the distance between two points on the circle.
What are some real-world applications of the equation of a circle? The equation of a circle has many real-world applications, including:
- Modeling the path of a projectile
- Designing lenses and mirrors
- Creating 2D and 3D shapes in computer graphics
- Calculating the area and circumference of a circle

Common Mistakes to Avoid

Not completing the square for both the $x$ and $y$ terms: Failing to complete the square for both the $x$ and $y$ terms can lead to incorrect solutions.
Not identifying the center and radius of the circle: Failing to identify the center and radius of the circle can lead to incorrect solutions.
Not using the correct formula for the radius: Using the wrong formula for the radius can lead to incorrect solutions.

Tips and Tricks

Use the equation of a circle to solve problems: The equation of a circle can be used to solve a wide range of problems, including those involving geometry, trigonometry, and calculus.
Practice completing the square: Completing the square is a crucial skill for solving problems involving the equation of a circle.
Use the correct formula for the radius: Using the correct formula for the radius is essential for solving problems involving the equation of a circle.

Real-World Examples

Designing a lens: The equation of a circle can be used to design a lens that focuses light onto a specific point.
Creating a 2D shape: The equation of a circle can be used to create a 2D shape that is symmetrical about its center.
Calculating the area of a circle: The equation of a circle can be used to calculate the area of a circle, which is essential for many real-world applications.

Mathematical Concepts

Completing the square: Completing the square is a crucial skill for solving problems involving the equation of a circle.
Standard form of the equation of a circle: The standard form of the 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.
Radius of a circle: The radius of a circle is given by $r = \sqrt{r^2}$ , where $r^2$ is the constant term on the right-hand side of the equation.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by James Stewart
[3] "Geometry" by Michael Artin

Additional Resources

Khan Academy: Completing the Square
MIT OpenCourseWare: Calculus
Wolfram Alpha: Equation of a Circle