In mathematics, circles are fascinating geometric shapes studied and admired for centuries. One of the fundamental properties of a circle is its radius, which is crucial for understanding its size, position, and various geometric properties. If given the equation of a circle, you can extract valuable information, including the radius. Let’s explore how to find the radius of a circle using the equation: x^2 + y^2 + 8x – 6y + 21 = 0.
Table of Contents
1. The General Equation of a Circle
The general equation of a circle is given by: (x – h)^2 + (y – k)^2 = r^2
Where (h, k) signifies the coordinates of the circle’s center, and r is the radius. We must convert the given equation into this standard form to find the radius.
2. Completing the Square
To do this, let’s complete the square for the given equation:
x^2 + y^2 + 8x – 6y + 21 = 0
Rearrange the terms by grouping the x-terms and y-terms:
(x^2 + 8x) + (y^2 – 6y) + 21 = 0
Now, complete the square for both the x and y terms. To complete the square for x-terms, we add (8/2)^2 = 16 to both sides, and for y-terms, we add (-6/2)^2 = 9 to both sides:
(x^2 + 8x + 16) + (y^2 – 6y + 9) + 21 = 16 + 9
3. Rewrite the Equation
Now, rewrite the equation in standard form:
(x + 4)^2 + (y – 3)^2 + 21 = 25
Subtract 21 from both sides to isolate the squared terms:
(x + 4)^2 + (y – 3)^2 = 25 – 21 (x + 4)^2 + (y – 3)^2 = 4
4. Compare with the Standard Form
Now that we have the equation in the standard form, we can see that the coordinates of the circle’s center are (-4, 3), and the radius squared is 4. To find the actual radius, take the square root of 4:
r = √4 = 2 units
Importance of the Circle Equation
Understanding how to work with the equation of a circle is essential in geometry, trigonometry, and many areas of science and engineering. Circles are prevalent in various real-world applications, from designing wheels and gears to calculating orbits in astronomy. The equation helps you determine important geometric properties, such as the center and radius, vital for various calculations and problem-solving tasks.
Conclusion
The radius of the circle with the equation x^2 + y^2 + 8x – 6y + 21 = 0 is 2 units. This information is essential for various applications in geometry, trigonometry, and other fields of mathematics and science. Understanding how to extract this information from the equation of a circle is a valuable skill for solving mathematical problems and real-world applications.
