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 x 2 + y 2 + 4 x − 8 y + 11 = 0 .
Complete the square for both x and y terms to rewrite the equation in standard form.
The equation becomes ( x + 2 ) 2 + ( y − 4 ) 2 = 9 .
The center of the circle is ( − 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, we complete the square for the x terms. We have x 2 + 4 x . To complete the square, we take half of the coefficient of the x term (which is 4), square it (which gives us ( 4/2 ) 2 = 2 2 = 4 ), and add and subtract it from the expression:


x 2 + 4 x = x 2 + 4 x + 4 − 4 = ( x + 2 ) 2 − 4

Complete the square for y terms Next, we complete the square for the y terms. We have y 2 − 8 y . To complete the square, we take half of the coefficient of the y term (which is -8), square it (which gives us ( − 8/2 ) 2 = ( − 4 ) 2 = 16 ), and add and subtract it from the expression:

y 2 − 8 y = y 2 − 8 y + 16 − 16 = ( y − 4 ) 2 − 16

Substitute back into the original equation Now, we substitute these expressions back into the original equation:

x 2 + y 2 + 4 x − 8 y + 11 = 0 becomes
(( x + 2 ) 2 − 4 ) + (( y − 4 ) 2 − 16 ) + 11 = 0

Simplify the equation Next, we simplify the equation:

( x + 2 ) 2 − 4 + ( y − 4 ) 2 − 16 + 11 = 0
( 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 Finally, we identify the center of the circle. Comparing ( x + 2 ) 2 + ( y − 4 ) 2 = 9 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 crucial in various fields. For instance, in GPS technology, determining the location of a device involves finding the intersection of circles. The standard form of a circle's equation helps in easily identifying the center and radius, which are essential for these calculations. Also, in architecture and engineering, knowing the center and radius is vital for designing circular structures or components.

Answered by GinnyAnswer | 2025-07-07