Finding The Radius Of A Circle The Equation $x^2+y^2+8x-6y+21=0$

Determining the radius of a circle from its equation is a fundamental concept in analytic geometry. In this comprehensive guide, we will delve into the process of finding the radius of a circle given its equation in the general form. Specifically, we'll tackle the equation $x^2 + y^2 + 8x - 6y + 21 = 0$. This exercise not only reinforces our understanding of circles but also showcases the power of algebraic manipulation in solving geometric problems. We'll break down the steps involved, ensuring clarity and fostering a deeper understanding of the underlying principles. Let's embark on this mathematical journey to unlock the secrets hidden within this equation.

The General Equation of a Circle and its Transformation

The general equation of a circle is expressed as $x^2 + y^2 + 2gx + 2fy + c = 0$, where (-g, -f) represents the center of the circle and the radius (r) is given by the formula $r = \sqrt{g^2 + f^2 - c}$. This equation is a powerful tool for describing circles in the Cartesian plane. However, it's not immediately obvious what the center and radius are just by looking at it. Therefore, we need a method to transform this general form into the standard form of a circle's equation. The standard form, $(x - h)^2 + (y - k)^2 = r^2$, provides a clear and concise representation of a circle's properties, with (h, k) denoting the center and r representing the radius. The process of converting from the general form to the standard form involves a technique called completing the square. This algebraic method allows us to rewrite quadratic expressions in a more manageable form, revealing the center and radius of the circle with ease. Understanding this transformation is key to solving a wide range of problems involving circles, making it a cornerstone of analytic geometry.

Completing the Square: A Step-by-Step Approach

Completing the square is a fundamental algebraic technique used to rewrite a quadratic expression into a perfect square trinomial plus a constant. This technique is crucial for transforming the general equation of a circle into its standard form. Let's break down the process step by step:

1. Group x and y terms: Begin by grouping the x terms together and the y terms together in the equation. For our equation, $x^2 + y^2 + 8x - 6y + 21 = 0$, this gives us $(x^2 + 8x) + (y^2 - 6y) + 21 = 0$.
2. Complete the square for x terms: To complete the square for the x terms, take half of the coefficient of the x term (which is 8), square it (resulting in 16), and add it to both sides of the equation. This gives us $(x^2 + 8x + 16) + (y^2 - 6y) + 21 = 16$.
3. Complete the square for y terms: Similarly, for the y terms, take half of the coefficient of the y term (which is -6), square it (resulting in 9), and add it to both sides of the equation. This gives us $(x^2 + 8x + 16) + (y^2 - 6y + 9) + 21 = 16 + 9$.
4. Rewrite as squared terms: Now, rewrite the x and y terms as perfect squares. The x terms become $(x + 4)^2$ and the y terms become $(y - 3)^2$. The equation now looks like $(x + 4)^2 + (y - 3)^2 + 21 = 25$.
5. Isolate the constant term: Move the constant term to the right side of the equation to obtain the standard form. Subtracting 21 from both sides gives us $(x + 4)^2 + (y - 3)^2 = 4$. This equation is now in the standard form of a circle's equation, making it easy to identify the center and radius.

By mastering the technique of completing the square, we can effectively transform the general equation of a circle into its standard form, unlocking the circle's key properties.

Identifying the Center and Radius

Once we have the equation in standard form, $(x - h)^2 + (y - k)^2 = r^2$, identifying the center and radius becomes a straightforward task. The center of the circle is represented by the coordinates (h, k), and the radius is simply the square root of the constant term on the right side of the equation, denoted as r. In our example, we transformed the general equation into the standard form $(x + 4)^2 + (y - 3)^2 = 4$. By comparing this to the standard form, we can readily identify the center and radius.

• Center: The x-coordinate of the center is the negation of the value inside the parenthesis with x, which is -4. The y-coordinate of the center is the value inside the parenthesis with y, which is 3. Therefore, the center of the circle is (-4, 3).
• Radius: The radius is the square root of the constant term on the right side of the equation, which is 4. Taking the square root of 4 gives us a radius of 2. Therefore, the radius of the circle is 2 units.

This direct relationship between the standard form of the equation and the circle's properties makes it a powerful tool for analyzing and understanding circles in the coordinate plane. By mastering the transformation to standard form, we gain immediate access to the circle's center and radius, simplifying various geometric calculations and problem-solving scenarios.

Applying the Concepts: Solving $x^2+y^2+8x-6y+21=0$

Now, let's apply the concepts we've discussed to solve the specific problem: finding the radius of the circle whose equation is $x^2 + y^2 + 8x - 6y + 21 = 0$. We'll follow the steps outlined in the "Completing the Square" section to transform the equation into standard form and then extract the radius.

1. Group x and y terms: $(x^2 + 8x) + (y^2 - 6y) + 21 = 0$
2. Complete the square for x terms: $(x^2 + 8x + 16) + (y^2 - 6y) + 21 = 16$
3. Complete the square for y terms: $(x^2 + 8x + 16) + (y^2 - 6y + 9) + 21 = 16 + 9$
4. Rewrite as squared terms: $(x + 4)^2 + (y - 3)^2 + 21 = 25$
5. Isolate the constant term: $(x + 4)^2 + (y - 3)^2 = 4$

We have successfully transformed the equation into standard form: $(x + 4)^2 + (y - 3)^2 = 4$. Comparing this to the standard form $(x - h)^2 + (y - k)^2 = r^2$, we can identify the center as (-4, 3) and $r^2$ as 4. To find the radius, we take the square root of 4, which gives us $r = 2$.

Therefore, the radius of the circle whose equation is $x^2 + y^2 + 8x - 6y + 21 = 0$ is 2 units. This step-by-step solution demonstrates the power of completing the square in unraveling the properties of a circle from its general equation.

Conclusion: Mastering Circle Equations

In conclusion, finding the radius of a circle from its general equation, such as $x^2 + y^2 + 8x - 6y + 21 = 0$, involves a systematic process of transforming the equation into standard form. The key technique is completing the square, which allows us to rewrite the equation in the form $(x - h)^2 + (y - k)^2 = r^2$. This standard form directly reveals the circle's center (h, k) and radius r. By following the steps of grouping terms, completing the square for both x and y variables, and isolating the constant term, we can successfully determine the radius.

In our specific example, we found that the radius of the circle described by the equation $x^2 + y^2 + 8x - 6y + 21 = 0$ is 2 units. This process not only provides the solution to this particular problem but also equips us with a powerful method for analyzing any circle equation in general form. Mastering these techniques enhances our understanding of analytic geometry and empowers us to solve a wide range of geometric problems involving circles. The ability to manipulate equations and extract meaningful information is a fundamental skill in mathematics, and this exercise serves as a valuable illustration of its importance.