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Sagot :
To arrange the circles represented by their equations in ascending order of their radius lengths, we'll follow these steps:
1. Convert each general form equation into the standard form of a circle equation [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex].
2. Identify the radius [tex]\( r \)[/tex] for each circle.
3. Compare the radii and sort the circles accordingly.
Let's start with the first equation:
1. Equation: [tex]\(x^2 + y^2 - 2x + y - 1 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 - 2x + y^2 + y = 1\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 2x \rightarrow (x - 1)^2 - 1 \][/tex]
[tex]\[ y^2 + y \rightarrow (y + \frac{1}{2})^2 - \frac{1}{4} \][/tex]
So, [tex]\( (x - 1)^2 - 1 + (y + \frac{1}{2})^2 - \frac{1}{4} = 1 \implies (x - 1)^2 + (y + \frac{1}{2})^2 = \frac{5}{4} \)[/tex]
- Radius [tex]\( r = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2} \)[/tex]
2. Equation: [tex]\(x^2 + y^2 - 4x + 4y - 10 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 - 4x + y^2 + 4y = 10\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 4x \rightarrow (x - 2)^2 - 4 \][/tex]
[tex]\[ y^2 + 4y \rightarrow (y + 2)^2 - 4 \][/tex]
So, [tex]\( (x - 2)^2 - 4 + (y + 2)^2 - 4 = 10 \implies (x - 2)^2 + (y + 2)^2 = 18 \)[/tex]
- Radius [tex]\( r = \sqrt{18} = 3\sqrt{2} \)[/tex]
3. Equation: [tex]\(x^2 + y^2 - 8x - 6y - 20 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 - 8x + y^2 - 6y = 20\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 8x \rightarrow (x - 4)^2 - 16 \][/tex]
[tex]\[ y^2 - 6y \rightarrow (y - 3)^2 - 9 \][/tex]
So, [tex]\( (x - 4)^2 - 16 + (y - 3)^2 - 9 = 20 \implies (x - 4)^2 + (y - 3)^2 = 45 \)[/tex]
- Radius [tex]\( r = \sqrt{45} = 3\sqrt{5} \)[/tex]
4. Equation: [tex]\(4x^2 + 4y^2 + 16x + 24y - 40 = 0\)[/tex]
- Divide by 4: [tex]\(x^2 + y^2 + 4x + 6y - 10 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 + 4x + y^2 + 6y = 10\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 + 4x \rightarrow (x + 2)^2 - 4 \][/tex]
[tex]\[ y^2 + 6y \rightarrow (y + 3)^2 - 9 \][/tex]
So, [tex]\( (x + 2)^2 - 4 + (y + 3)^2 - 9 = 10 \implies (x + 2)^2 + (y + 3)^2 = 23 \)[/tex]
- Radius [tex]\( r = \sqrt{23} \)[/tex]
5. Equation: [tex]\(5x^2 + 5y^2 - 20x + 30y + 40 = 0\)[/tex]
- Divide by 5: [tex]\(x^2 + y^2 - 4x + 6y + 8 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 - 4x + y^2 + 6y = -8\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 4x \rightarrow (x - 2)^2 - 4 \][/tex]
[tex]\[ y^2 + 6y \rightarrow (y + 3)^2 - 9 \][/tex]
So, [tex]\( (x - 2)^2 - 4 + (y + 3)^2 - 9 = -8 \implies (x - 2)^2 + (y + 3)^2 = 5 \)[/tex]
- Radius [tex]\( r = \sqrt{5} \)[/tex]
6. Equation: [tex]\(2x^2 + 2y^2 - 28x - 32y - 8 = 0\)[/tex]
- Divide by 2: [tex]\(x^2 + y^2 - 14x - 16y - 4 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 - 14x + y^2 - 16y = 4\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 14x \rightarrow (x - 7)^2 - 49 \][/tex]
[tex]\[ y^2 - 16y \rightarrow (y - 8)^2 - 64 \][/tex]
So, [tex]\( (x - 7)^2 - 49 + (y - 8)^2 - 64 = 4 \implies (x - 7)^2 + (y - 8)^2 = 117 \)[/tex]
- Radius [tex]\( r = \sqrt{117} = 3\sqrt{13} \)[/tex]
7. Equation: [tex]\(x^2 + y^2 + 12x - 2y - 9 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 + 12x + y^2 - 2y = 9\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 + 12x \rightarrow (x + 6)^2 - 36 \][/tex]
[tex]\[ y^2 - 2y \rightarrow (y - 1)^2 - 1 \][/tex]
So, [tex]\( (x + 6)^2 - 36 + (y - 1)^2 - 1 = 9 \implies (x + 6)^2 + (y - 1)^2 = 46 \)[/tex]
- Radius [tex]\( r = \sqrt{46} \)[/tex]
Now, let's arrange these radii in ascending order (smallest to largest):
1. [tex]\(\frac{\sqrt{5}}{2}\)[/tex]
2. [tex]\(\sqrt{5}\)[/tex]
3. [tex]\(\sqrt{23}\)[/tex]
4. [tex]\(\sqrt{46}\)[/tex]
5. [tex]\(3\sqrt{2}\)[/tex]
6. [tex]\(3\sqrt{5}\)[/tex]
7. [tex]\(3\sqrt{13}\)[/tex]
Therefore, the equations ordered by the radius (smallest to largest) are:
1. [tex]\(x^2 + y^2 - 2x + y - 1 = 0\)[/tex]
2. [tex]\(5x^2 + 5y^2 - 20x + 30y + 40 = 0\)[/tex]
3. [tex]\(4x^2 + 4y^2 + 16x + 24y - 40 = 0\)[/tex]
4. [tex]\(x^2 + y^2 + 12x - 2y - 9 = 0\)[/tex]
5. [tex]\(x^2 + y^2 - 4x + 4y - 10 = 0\)[/tex]
6. [tex]\(x^2 + y^2 - 8x - 6y - 20 = 0\)[/tex]
7. [tex]\(2x^2 + 2y^2 - 28x - 32y - 8 = 0\)[/tex]
1. Convert each general form equation into the standard form of a circle equation [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex].
2. Identify the radius [tex]\( r \)[/tex] for each circle.
3. Compare the radii and sort the circles accordingly.
Let's start with the first equation:
1. Equation: [tex]\(x^2 + y^2 - 2x + y - 1 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 - 2x + y^2 + y = 1\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 2x \rightarrow (x - 1)^2 - 1 \][/tex]
[tex]\[ y^2 + y \rightarrow (y + \frac{1}{2})^2 - \frac{1}{4} \][/tex]
So, [tex]\( (x - 1)^2 - 1 + (y + \frac{1}{2})^2 - \frac{1}{4} = 1 \implies (x - 1)^2 + (y + \frac{1}{2})^2 = \frac{5}{4} \)[/tex]
- Radius [tex]\( r = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2} \)[/tex]
2. Equation: [tex]\(x^2 + y^2 - 4x + 4y - 10 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 - 4x + y^2 + 4y = 10\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 4x \rightarrow (x - 2)^2 - 4 \][/tex]
[tex]\[ y^2 + 4y \rightarrow (y + 2)^2 - 4 \][/tex]
So, [tex]\( (x - 2)^2 - 4 + (y + 2)^2 - 4 = 10 \implies (x - 2)^2 + (y + 2)^2 = 18 \)[/tex]
- Radius [tex]\( r = \sqrt{18} = 3\sqrt{2} \)[/tex]
3. Equation: [tex]\(x^2 + y^2 - 8x - 6y - 20 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 - 8x + y^2 - 6y = 20\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 8x \rightarrow (x - 4)^2 - 16 \][/tex]
[tex]\[ y^2 - 6y \rightarrow (y - 3)^2 - 9 \][/tex]
So, [tex]\( (x - 4)^2 - 16 + (y - 3)^2 - 9 = 20 \implies (x - 4)^2 + (y - 3)^2 = 45 \)[/tex]
- Radius [tex]\( r = \sqrt{45} = 3\sqrt{5} \)[/tex]
4. Equation: [tex]\(4x^2 + 4y^2 + 16x + 24y - 40 = 0\)[/tex]
- Divide by 4: [tex]\(x^2 + y^2 + 4x + 6y - 10 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 + 4x + y^2 + 6y = 10\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 + 4x \rightarrow (x + 2)^2 - 4 \][/tex]
[tex]\[ y^2 + 6y \rightarrow (y + 3)^2 - 9 \][/tex]
So, [tex]\( (x + 2)^2 - 4 + (y + 3)^2 - 9 = 10 \implies (x + 2)^2 + (y + 3)^2 = 23 \)[/tex]
- Radius [tex]\( r = \sqrt{23} \)[/tex]
5. Equation: [tex]\(5x^2 + 5y^2 - 20x + 30y + 40 = 0\)[/tex]
- Divide by 5: [tex]\(x^2 + y^2 - 4x + 6y + 8 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 - 4x + y^2 + 6y = -8\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 4x \rightarrow (x - 2)^2 - 4 \][/tex]
[tex]\[ y^2 + 6y \rightarrow (y + 3)^2 - 9 \][/tex]
So, [tex]\( (x - 2)^2 - 4 + (y + 3)^2 - 9 = -8 \implies (x - 2)^2 + (y + 3)^2 = 5 \)[/tex]
- Radius [tex]\( r = \sqrt{5} \)[/tex]
6. Equation: [tex]\(2x^2 + 2y^2 - 28x - 32y - 8 = 0\)[/tex]
- Divide by 2: [tex]\(x^2 + y^2 - 14x - 16y - 4 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 - 14x + y^2 - 16y = 4\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 14x \rightarrow (x - 7)^2 - 49 \][/tex]
[tex]\[ y^2 - 16y \rightarrow (y - 8)^2 - 64 \][/tex]
So, [tex]\( (x - 7)^2 - 49 + (y - 8)^2 - 64 = 4 \implies (x - 7)^2 + (y - 8)^2 = 117 \)[/tex]
- Radius [tex]\( r = \sqrt{117} = 3\sqrt{13} \)[/tex]
7. Equation: [tex]\(x^2 + y^2 + 12x - 2y - 9 = 0\)[/tex]
- Group [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms: [tex]\(x^2 + 12x + y^2 - 2y = 9\)[/tex]
- Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 + 12x \rightarrow (x + 6)^2 - 36 \][/tex]
[tex]\[ y^2 - 2y \rightarrow (y - 1)^2 - 1 \][/tex]
So, [tex]\( (x + 6)^2 - 36 + (y - 1)^2 - 1 = 9 \implies (x + 6)^2 + (y - 1)^2 = 46 \)[/tex]
- Radius [tex]\( r = \sqrt{46} \)[/tex]
Now, let's arrange these radii in ascending order (smallest to largest):
1. [tex]\(\frac{\sqrt{5}}{2}\)[/tex]
2. [tex]\(\sqrt{5}\)[/tex]
3. [tex]\(\sqrt{23}\)[/tex]
4. [tex]\(\sqrt{46}\)[/tex]
5. [tex]\(3\sqrt{2}\)[/tex]
6. [tex]\(3\sqrt{5}\)[/tex]
7. [tex]\(3\sqrt{13}\)[/tex]
Therefore, the equations ordered by the radius (smallest to largest) are:
1. [tex]\(x^2 + y^2 - 2x + y - 1 = 0\)[/tex]
2. [tex]\(5x^2 + 5y^2 - 20x + 30y + 40 = 0\)[/tex]
3. [tex]\(4x^2 + 4y^2 + 16x + 24y - 40 = 0\)[/tex]
4. [tex]\(x^2 + y^2 + 12x - 2y - 9 = 0\)[/tex]
5. [tex]\(x^2 + y^2 - 4x + 4y - 10 = 0\)[/tex]
6. [tex]\(x^2 + y^2 - 8x - 6y - 20 = 0\)[/tex]
7. [tex]\(2x^2 + 2y^2 - 28x - 32y - 8 = 0\)[/tex]
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