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. $(4,-8)$
Answer (1)
Rewrite the given equation x 2 + y 2 + 4 x − 8 y + 11 = 0 in the standard form of a circle's equation.
Complete the square for the x terms: x 2 + 4 x = ( x + 2 ) 2 − 4 .
Complete the square for the y terms: y 2 − 8 y = ( y − 4 ) 2 − 16 .
Identify the center of the circle from the standard form: ( − 2 , 4 ) .
Explanation
Analyze the problem and rewrite in standard form 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 of a circle, which is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center 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 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 From the standard form ( x + 2 ) 2 + ( y − 4 ) 2 = 9 , we can identify the center of the circle as ( − 2 , 4 ) .
State the final answer Therefore, the center of the circle is ( − 2 , 4 ) .
Examples
Understanding the equation of a circle is crucial in various fields. For example, in GPS technology, your location is determined by finding the intersection of circles from multiple satellites. Each satellite sends a signal indicating your distance from it, which defines a circle (or sphere in 3D). The center of each circle is the satellite's location, and the radius is the distance to you. By solving the system of equations representing these circles, your precise location can be determined. This principle extends to many other applications, such as radar systems, computer graphics, and even art, where circles and their properties are fundamental.
