Mrs. Culland is finding the center of a circle whose equation is $x^2+y^2+6 x+4 y-3=0$ by completing the square. Her work is shown.
$x^2+y^2+6 x+4 y-3=0$
$x^2+6 x+y^2+4 y-3=0$
$(x^2+6 x)+(y^2+4 y)=3$
$(x^2+6 x+9)+(y^2+4 y+4)=3+\underline{9}+4$
Which completes the work correctly?
A. $(x-3)^2+(y-2)^2=4^2$, so the center is $(3,2)$.
B. $(x+3)^2+(y+2)^2=4^2$, so the center is $(3,2)$.
C. $(x-3)^2+(y-2)^2=4^2$, so the center is $(-3,-2)$.
D. $(x+3)^2+(y+2)^2=4^2$, so the center is $(-3,-2)$.
Answer (1)
Rearrange the given equation to group x and y terms: x 2 + 6 x + y 2 + 4 y = 3 .
Complete the square for both x and y terms: ( x 2 + 6 x + 9 ) + ( y 2 + 4 y + 4 ) = 3 + 9 + 4 .
Rewrite the equation in standard circle form: ( x + 3 ) 2 + ( y + 2 ) 2 = 4 2 .
Identify the center of the circle as ( − 3 , − 2 ) .
Explanation
Analyze the problem We are given the equation of a circle: x 2 + y 2 + 6 x + 4 y − 3 = 0 . Our goal is to find the center of this circle by completing the square.
Rearrange the equation First, we rearrange the equation to group the x and y terms together: x 2 + 6 x + y 2 + 4 y = 3
Complete the square Next, we complete the square for the x terms. To complete the square for x 2 + 6 x , we need to add ( 2 6 ) 2 = 3 2 = 9 to both sides of the equation. Similarly, to complete the square for y 2 + 4 y , we need to add ( 2 4 ) 2 = 2 2 = 4 to both sides of the equation.
Rewrite the equation So, we have: ( x 2 + 6 x + 9 ) + ( y 2 + 4 y + 4 ) = 3 + 9 + 4 ( x + 3 ) 2 + ( y + 2 ) 2 = 16 ( x + 3 ) 2 + ( y + 2 ) 2 = 4 2
Identify the center and radius The equation of a circle in standard form is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center of the circle and r is the radius. In our case, we have ( x + 3 ) 2 + ( y + 2 ) 2 = 4 2 , which can be written as ( x − ( − 3 ) ) 2 + ( y − ( − 2 ) ) 2 = 4 2 . Therefore, the center of the circle is ( − 3 , − 2 ) and the radius is 4 .
Examples
Understanding the equation of a circle is useful in many real-world applications. For example, civil engineers use it when designing circular structures such as tunnels or roundabouts. Architects use it when designing curved windows or domes. In navigation, the equation of a circle is used to determine the range of a radar or sonar system. By knowing the center and radius of a circle, we can accurately map and analyze circular shapes in various fields.
