Drag the tiles to the boxes to form correct pairs. Not all tiles will be used.

Match the equations that represent concentric circles. Concentric circles are circles with the same center.

[tex]\[
\begin{array}{cc}
3x^2 + 3y^2 - 18x + 6y + 6 = 0 & 4x^2 + 4y^2 + 8x - 40y - 164 = 0 \\
x^2 + y^2 + 4x + 8y + 4 = 0 & x^2 + y^2 + 2x + 8y - 40 = 0 \\
\end{array}
\][/tex]

[tex]\[
5x^2 + 5y^2 + 10x + 20y + 5 = 0
\][/tex]

[tex]\[
\begin{array}{l}
x^2 + y^2 - 6x + 2y + 8 = 0 \\
2x^2 + 2y^2 + 4x + 16y - 10 = 0 \\
5x^2 + 5y^2 + 10x - 50y - 200 = 0 \\
\end{array}
\][/tex]



Answer :

To determine which of the given equations represent concentric circles, we need to focus on identifying circles that share the same center. Here is a step-by-step solution:

1. Identify the Circle's Equations:
- Each equation given can be represented in the standard form of a circle's equation, [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], after some manipulation.

2. Group the Equations Based on Similar Coefficients:
- Since the equations are in the form [tex]\(Ax^2 + Ay^2 + Dx + Ey + F = 0\)[/tex], we will compare the transformed forms to find equations that share centers, [tex]\((h, k)\)[/tex].

3. Equation Pairs:

- First Pair:
- [tex]\(3x^2 + 3y^2 - 18x + 6y + 6 = 0\)[/tex]
- [tex]\(x^2 + y^2 - 6x + 2y + 8 = 0\)[/tex]
- Transform the first equation to the standard form:
- Divide by 3: [tex]\(x^2 + y^2 - 6x + 2y + 2 = 0\)[/tex]
- Compare [tex]\((h, k)\)[/tex]:
- Both equations have the same center [tex]\((h, k) = (3, -1)\)[/tex].

- Second Pair:
- [tex]\(4x^2 + 4y^2 + 8x - 40y - 164 = 0\)[/tex]
- [tex]\(5x^2 + 5y^2 + 10x - 50y - 200 = 0\)[/tex]
- For [tex]\(4x^2 + 4y^2 + 8x - 40y - 164 = 0\)[/tex]:
- Divide by 4: [tex]\(x^2 + y^2 + 2x - 10y - 41 = 0\)[/tex]
- For [tex]\(5x^2 + 5y^2 + 10x - 50y - 200 = 0\)[/tex]:
- Divide by 5: [tex]\(x^2 + y^2 + 2x - 10y - 40 = 0\)[/tex]
- Both have the same center [tex]\((h, k) = (-1, 5)\)[/tex].

- Third Pair:
- [tex]\(x^2 + y^2 + 2x + 8y - 40 = 0\)[/tex]
- [tex]\(2x^2 + 2y^2 + 4x + 16y - 10 = 0\)[/tex]
- For [tex]\(2x^2 + 2y^2 + 4x + 16y - 10 = 0\)[/tex]:
- Divide by 2: [tex]\(x^2 + y^2 + 2x + 8y - 5 = 0\)[/tex]
- Both have the same center [tex]\((h, k) = (-1, -4)\)[/tex].

4. Final Pairs of Concentric Circles:

- [tex]\((3x^2 + 3y^2 - 18x + 6y + 6 = 0,\; x^2 + y^2 - 6x + 2y + 8 = 0)\)[/tex]
- [tex]\((4x^2 + 4y^2 + 8x - 40y - 164 = 0,\; 5x^2 + 5y^2 + 10x - 50y - 200 = 0)\)[/tex]
- [tex]\((x^2 + y^2 + 2x + 8y - 40 = 0,\; 2x^2 + 2y^2 + 4x + 16y - 10 = 0)\)[/tex]

Thus, the concentric circle pairs should be matched as follows:
- [tex]\(3x^2 + 3y^2 - 18x + 6y + 6 = 0\)[/tex] with [tex]\(x^2 + y^2 - 6x + 2y + 8 = 0\)[/tex]
- [tex]\(4x^2 + 4y^2 + 8x - 40y - 164 = 0\)[/tex] with [tex]\(5x^2 + 5y^2 + 10x - 50y - 200 = 0\)[/tex]
- [tex]\(x^2 + y^2 + 2x + 8y - 40 = 0\)[/tex] with [tex]\(2x^2 + 2y^2 + 4x + 16y - 10 = 0\)[/tex]