What is the center of a circle whose equation is $x^2+y^2+4 x-8 y+11=0$?
A. $(-2,4)$
B. $(-4,8)$
C. $(2,-4)$
D. $(4,-8)$

We are given the equation of a circle and aim to find its center.
We complete the square for both x and y terms to rewrite the equation in standard form.
The equation is transformed to ( x + 2 ) 2 + ( y − 4 ) 2 = 9 .
The center of the circle is identified as ( − 2 , 4 ) ​ .

Explanation

Analyze the problem and rewrite the equation. We are given the equation of a circle: x 2 + y 2 + 4 x − 8 y + 11 = 0 . Our goal is to find the center of this circle. To do this, we will rewrite the equation in the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) represents the center of the circle and r is the radius.

Complete the square for x terms. First, let's complete the square for the x terms. We have x 2 + 4 x . To complete the square, we need to add and subtract ( 2 4 ​ ) 2 = 2 2 = 4 . So, x 2 + 4 x = ( x 2 + 4 x + 4 ) − 4 = ( x + 2 ) 2 − 4 .

Complete the square for y terms. Next, let's complete the square for the y terms. We have y 2 − 8 y . To complete the square, we need to add and subtract ( 2 − 8 ​ ) 2 = ( − 4 ) 2 = 16 . So, y 2 − 8 y = ( y 2 − 8 y + 16 ) − 16 = ( y − 4 ) 2 − 16 .

Substitute back into the original equation. Now, substitute these back into the original equation: ( x + 2 ) 2 − 4 + ( y − 4 ) 2 − 16 + 11 = 0 .

Simplify the equation. Simplify the equation: ( x + 2 ) 2 + ( y − 4 ) 2 − 4 − 16 + 11 = 0 ( x + 2 ) 2 + ( y − 4 ) 2 − 9 = 0 ( x + 2 ) 2 + ( y − 4 ) 2 = 9

Identify the center of the circle. Now we can identify the center of the circle. Comparing this equation with the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , we have h = − 2 and k = 4 . Therefore, the center of the circle is ( − 2 , 4 ) .


Examples
Understanding the equation of a circle is very useful in many real-world applications. For example, civil engineers use it when designing circular structures like tunnels or roundabouts. By knowing the equation of a circle, they can accurately determine the center and radius, which are crucial for construction and ensuring structural integrity. Also, in computer graphics, circles are fundamental shapes, and their equations are used to draw and manipulate circular objects on the screen. Knowing the center and radius allows for precise placement and scaling of these objects, which is essential for creating visually appealing and accurate graphics.

Answered by GinnyAnswer | 2025-07-08