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.
$x^2+y^2+6 x+4 y-3=0$
$x^2+6 x+y^2+4 y-3=0$
$(x^2+6 x)+(y^2+4 y)=3$
$(x^2+6 x+9)+(y^2+4 y+4)=3+9+4$
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 (2)
Group the x and y terms and move the constant to the right side: x 2 + 6 x + y 2 + 4 y = 3 .
Complete the square for x and y terms: ( x 2 + 6 x + 9 ) + ( y 2 + 4 y + 4 ) = 3 + 9 + 4 .
Rewrite the equation in standard form: ( x + 3 ) 2 + ( y + 2 ) 2 = 16 , which is ( x + 3 ) 2 + ( y + 2 ) 2 = 4 2 .
Identify the center of the circle: ( − 3 , − 2 ) .
Explanation
Analyze the problem We are given the equation of a circle: x 2 + y 2 + 6 x + 4 y − 3 = 0 . Mrs. Culland is trying to find the center of the circle by completing the square. Let's follow her steps and complete the square correctly.
Group x and y terms First, we group the x terms and y terms together and move the constant to the right side of the equation: 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. So, we have: ( x 2 + 6 x + 9 ) + ( y 2 + 4 y + 4 ) = 3 + 9 + 4
Rewrite in standard form Now, we can rewrite the equation as: ( x + 3 ) 2 + ( y + 2 ) 2 = 16
Express radius as a square Since 16 = 4 2 , the equation is: ( x + 3 ) 2 + ( y + 2 ) 2 = 4 2
Identify the center The standard form of a circle's equation is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius. In our case, h = − 3 and k = − 2 . Therefore, the center of the circle is ( − 3 , − 2 ) .
State the final answer The correct completion of the work is ( x + 3 ) 2 + ( y + 2 ) 2 = 4 2 , so the center is ( − 3 , − 2 ) .
Examples
Completing the square is a useful technique in various fields, such as physics and engineering. For example, when analyzing the motion of a projectile, completing the square can help determine the maximum height reached by the projectile. Similarly, in electrical engineering, it can be used to analyze circuits and determine the resonant frequency. This technique provides a systematic way to rewrite quadratic expressions into a more manageable form, making it easier to solve problems and gain insights into the underlying phenomena.
The center of the circle given the equation x 2 + y 2 + 6 x + 4 y − 3 = 0 is found to be ( − 3 , − 2 ) by completing the square. The correct answer is option D.
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