Find The Center Of The Circle X² + Y² + 4x - 8y + 11 = 0 A Step-by-Step Guide
Determining the center of a circle given its equation is a fundamental concept in analytic geometry. This article will guide you through the process of finding the center of a circle, using the specific equation x² + y² + 4x - 8y + 11 = 0 as an example. We'll break down the steps, explain the underlying principles, and provide a clear, easy-to-follow solution.
Understanding the Standard Equation of a Circle
To effectively find the center of a circle, it's crucial to understand the standard equation of a circle. The standard form equation allows us to easily identify the circle's center and radius, which are key characteristics of any circle. The standard equation of a circle with center (h, k) and radius r is given by:
(x - h)² + (y - k)² = r²
In this equation:
- (h, k) represents the coordinates of the center of the circle.
- r represents the radius of the circle.
The equation x² + y² + 4x - 8y + 11 = 0 is in the general form of a circle's equation. To find the center, we need to convert it to the standard form. This involves a process called completing the square. The general form of a circle's equation is given by:
x² + y² + Dx + Ey + F = 0
Where D, E, and F are constants. Our goal is to transform the given equation into the standard form, which will directly reveal the center (h, k) and the radius r. Understanding the standard form and how it relates to the general form is the first critical step in finding the center of a circle.
Completing the Square: Transforming the Equation
The core technique for finding the center of the circle from the given equation x² + y² + 4x - 8y + 11 = 0 is completing the square. This method allows us to rewrite quadratic expressions in a form that reveals the center and radius of the circle. Completing the square involves manipulating the equation to create perfect square trinomials for both the x and y terms. Let's break down the steps involved in this process.
Step 1: Group the x and y terms
First, rearrange the equation to group the x terms together and the y terms together:
(x² + 4x) + (y² - 8y) = -11
We've moved the constant term to the right side of the equation to prepare for completing the square.
Step 2: Complete the square for the x terms
To complete the square for the x terms (x² + 4x), we need to add and subtract the square of half the coefficient of the x term. The coefficient of x is 4, so half of it is 2, and the square of 2 is 4. Thus, we add and subtract 4:
(x² + 4x + 4) - 4
Now, the expression (x² + 4x + 4) is a perfect square trinomial, which can be factored as (x + 2)².
Step 3: Complete the square for the y terms
Similarly, for the y terms (y² - 8y), we need to add and subtract the square of half the coefficient of the y term. The coefficient of y is -8, so half of it is -4, and the square of -4 is 16. Thus, we add and subtract 16:
(y² - 8y + 16) - 16
The expression (y² - 8y + 16) is also a perfect square trinomial, which can be factored as (y - 4)².
Step 4: Rewrite the equation
Now, substitute the completed square expressions back into the equation:
(x² + 4x + 4) - 4 + (y² - 8y + 16) - 16 = -11
Rewrite the perfect square trinomials and move the constants to the right side:
(x + 2)² + (y - 4)² = -11 + 4 + 16
Simplify the right side:
(x + 2)² + (y - 4)² = 9
Now the equation is in the standard form of a circle's equation, (x - h)² + (y - k)² = r². By completing the square, we have successfully transformed the given equation into a form that allows us to easily identify the center and radius of the circle. This method is a powerful tool in analytic geometry for working with conic sections.
Identifying the Center: Reading from the Standard Form
After completing the square, we've transformed the equation x² + y² + 4x - 8y + 11 = 0 into the standard form: (x + 2)² + (y - 4)² = 9. The standard form of a circle's equation, (x - h)² + (y - k)² = r², directly reveals the center (h, k) and the radius r. In this section, we'll focus on extracting the coordinates of the center from the standard form equation we derived.
Comparing with the Standard Equation
By comparing our transformed equation, (x + 2)² + (y - 4)² = 9, with the general standard form, (x - h)² + (y - k)² = r², we can directly identify the values of h and k, which represent the x and y coordinates of the center, respectively.
- (x + 2)² can be rewritten as (x - (-2))², so h = -2.
- (y - 4)² remains as is, so k = 4.
Therefore, the center of the circle is at the point (-2, 4). It's important to note the signs when extracting the coordinates. A term like (x + 2) corresponds to an x-coordinate of -2, and a term like (y - 4) corresponds to a y-coordinate of 4.
Determining the Radius
While our primary focus is on finding the center, the standard form also provides the radius. The right side of the equation, 9, represents r², where r is the radius. To find the radius, we take the square root of 9:
r = √9 = 3
So, the radius of the circle is 3 units. However, for this problem, we were specifically asked to find the center, which we have determined to be (-2, 4). Identifying the center from the standard form is a straightforward process once the equation is in the correct format. This skill is fundamental in various geometric and algebraic applications.
The Solution: Putting it All Together
Having walked through the process step-by-step, let's consolidate our findings. The original equation given was x² + y² + 4x - 8y + 11 = 0. Our goal was to find the center of the circle represented by this equation. We achieved this by employing the technique of completing the square, which allowed us to transform the given equation into the standard form of a circle's equation.
Recap of the Steps
- Grouped x and y terms: We rearranged the equation to group the x terms (x² + 4x) and the y terms (y² - 8y) together, moving the constant term to the right side.
- Completed the square: We completed the square for both the x and y terms by adding and subtracting the square of half the coefficient of the x and y terms, respectively. This allowed us to rewrite the expressions as perfect square trinomials.
- Rewrote the equation in standard form: We substituted the completed square expressions back into the equation and simplified to obtain the standard form: (x + 2)² + (y - 4)² = 9.
- Identified the center: By comparing the standard form equation with the general form (x - h)² + (y - k)² = r², we identified the center as (-2, 4).
The Answer
Therefore, the center of the circle whose equation is x² + y² + 4x - 8y + 11 = 0 is (-2, 4). This corresponds to option A in the multiple-choice options provided.
Conclusion: Mastering Circle Equations
In conclusion, finding the center of a circle from its general equation involves a systematic approach, with completing the square being the key technique. This method allows us to transform the equation into the standard form, which directly reveals the center and radius of the circle. Understanding the standard equation of a circle and mastering the process of completing the square are essential skills in analytic geometry.
This article has provided a detailed, step-by-step guide to solving the specific problem of finding the center of the circle given by the equation x² + y² + 4x - 8y + 11 = 0. By following these steps, you can confidently tackle similar problems and deepen your understanding of circles and their equations. Furthermore, this knowledge can be applied to various other geometric problems and concepts, making it a valuable skill to acquire.
