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Sagot :
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]
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]
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