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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.



$\begin{array}{l}

x^2+y^2+6 x+4 y-3=0 \\

x^2+6 x+y^2+4 y-3=0 \\

\left(x^2+6 x\right)+\left(y^2+4 y\right)=3 \\

\left(x^2+6 x+9\right)+\left(y^2+4 y+4\right)=3+9+4

\end{array}$



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 $(-3,-2)$.

Verified Answer :

  • Group the \(x\) and \(y\) terms of the circle equation \(x^2 + y^2 + 6x + 4y - 3 = 0\).
  • Add the necessary constants (\(9\) and \(4\)) to both sides to complete the squares for \(x\) and \(y\).
  • Rewrite the equation in the standard form \((x + 3)^2 + (y + 2)^2 = 4^2\).
  • Identify the center \((h, k)\) as \((-3, -2)\) from the standard form \((x - h)^2 + (y - k)^2 = r^2\).
  • The correct option is: \((x + 3)^2 + (y + 2)^2 = 4^2\), so the center is \((-3, -2)\).

Explanation

  1. Analyze the given circle equation and complete the square.

The objective is to find the center of the circle represented by the equation \(x^2 + y^2 + 6x + 4y - 3 = 0\) by completing the square. The equation for a circle in standard form is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. We start with the given equation and group the \(x\) and \(y\) terms:
\[x^2 + 6x + y^2 + 4y = 3\]
To complete the square for \(x^2 + 6x\), we take half of the coefficient of \(x\) (which is \(6\div 2 = 3\)), and square it (\(3^2 = 9\)). Adding 9 to both sides of the equation gives \((x^2 + 6x + 9)\).
Similarly, to complete the square for \(y^2 + 4y\), we take half of the coefficient of \(y\) (which is \(4\div 2 = 2\)), and square it (\(2^2 = 4\)). Adding 4 to both sides of the equation gives \((y^2 + 4y + 4)\).
Adding these to the equation leads to:
\[(x^2 + 6x + 9) + (y^2 + 4y + 4) = 3 + 9 + 4\]

  1. Rewrite the equation in standard form and identify the center.

Now, simplify the equation. The left side can be written as the sum of two perfect square binomials, and the right side is the sum of the constants:
\[(x + 3)^2 + (y + 2)^2 = 16\]
Since \(16 = 4^2\), the equation becomes:
\[(x + 3)^2 + (y + 2)^2 = 4^2\]
Comparing this to the standard circle equation form \((x - h)^2 + (y - k)^2 = r^2\), we have:
\(x - h = x + 3 \implies h = -3\)
\(y - k = y + 2 \implies k = -2\)
Therefore, the center of the circle \((h, k)\) is \((-3, -2)\). Comparing this to the given options, the final choice is \((x + 3)^2 + (y + 2)^2 = 4^2\), so the center is \((-3, -2)\).

Examples

Understanding how to complete the square is a fundamental skill in geometry for analyzing circles. For instance, if an architect needs to determine the center of a circular architectural feature on a coordinate-based blueprint to ensure it is centered on a specific support pillar, they would use this algebraic technique to transform a general equation into a readable center-radius format. This skill is also vital in fields like physics or data science, where one might need to identify the parameters of a circular cluster or a spherical trajectory in a 2D or 3D coordinate system.

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