Answer :
To determine which equations represent circles with the smallest and largest radii, we first need to rewrite each equation in the standard form of a circle's equation: [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex]. This is accomplished by completing the square.
Let's complete the square for each equation:
### Equation 1:
[tex]\[ 2x^2 + 2y^2 + 16x - 4y + 30 = 0 \][/tex]
1. Divide by 2 to simplify:
[tex]\[ x^2 + y^2 + 8x - 2y + 15 = 0 \][/tex]
2. Move the constant term to the right side:
[tex]\[ x^2 + y^2 + 8x - 2y = -15 \][/tex]
3. Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 + 8x + 16 + y^2 - 2y + 1 = -15 + 16 + 1 \][/tex]
4. Factor the perfect squares and solve for the right side:
[tex]\[ (x + 4)^2 + (y - 1)^2 = 2 \][/tex]
Here, the radius [tex]\(r_1 = \sqrt{2}\)[/tex].
### Equation 2:
[tex]\[ x^2 + y^2 + 6x - 4y - 20 = 0 \][/tex]
1. Move the constant term to the right side:
[tex]\[ x^2 + y^2 + 6x - 4y = 20 \][/tex]
2. Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 + 6x + 9 + y^2 - 4y + 4 = 20 + 9 + 4 \][/tex]
3. Factor the perfect squares and solve for the right side:
[tex]\[ (x + 3)^2 + (y - 2)^2 = 33 \][/tex]
Here, the radius [tex]\(r_2 = \sqrt{33}\)[/tex].
### Equation 3:
[tex]\[ 4x^2 + 4y^2 - 16x - 24y + 51 = 0 \][/tex]
1. Divide by 4 to simplify:
[tex]\[ x^2 + y^2 - 4x - 6y + \frac{51}{4} = 0 \][/tex]
2. Move the constant term to the right side:
[tex]\[ x^2 + y^2 - 4x - 6y = -\frac{51}{4} \][/tex]
3. Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 4x + 4 + y^2 - 6y + 9 = -\frac{51}{4} + 4 + 9 \][/tex]
4. Calculate the right side:
[tex]\[ -\frac{51}{4} + 4 + 9 = -\frac{51}{4} + \frac{16}{4} + \frac{36}{4} = -\frac{51 - 16 - 36}{4} = \frac{1}{4} \][/tex]
5. Factor the perfect squares:
[tex]\[ (x - 2)^2 + (y - 3)^2 = \frac{1}{4} \][/tex]
Here, the radius [tex]\(r_3 = \sqrt{\frac{1}{4}} = \frac{1}{2}\)[/tex].
### Summary:
1. Radius for Equation 1: [tex]\( r_1 = \sqrt{2} \)[/tex]
2. Radius for Equation 2: [tex]\( r_2 = \sqrt{33} \)[/tex]
3. Radius for Equation 3: [tex]\( r_3 = \frac{1}{2} \)[/tex]
Smallest radius: [tex]\( \frac{1}{2} \)[/tex] (Equation 3: [tex]\( 4x^2 + 4y^2 - 16x - 24y + 51 = 0 \)[/tex])
Largest radius: [tex]\( \sqrt{33} \)[/tex] (Equation 2: [tex]\( x^2 + y^2 + 6x - 4y - 20 = 0 \)[/tex])
The table should be completed as follows:
| Radius | Equation |
|---------------|------------------------------------------------|
| Smallest | [tex]\( 4 x^2 + 4 y^2 - 16 x - 24 y + 51 = 0 \)[/tex] |
| Largest | [tex]\( x^2 + y^2 + 6 x - 4 y - 20 = 0 \)[/tex] |
Let's complete the square for each equation:
### Equation 1:
[tex]\[ 2x^2 + 2y^2 + 16x - 4y + 30 = 0 \][/tex]
1. Divide by 2 to simplify:
[tex]\[ x^2 + y^2 + 8x - 2y + 15 = 0 \][/tex]
2. Move the constant term to the right side:
[tex]\[ x^2 + y^2 + 8x - 2y = -15 \][/tex]
3. Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 + 8x + 16 + y^2 - 2y + 1 = -15 + 16 + 1 \][/tex]
4. Factor the perfect squares and solve for the right side:
[tex]\[ (x + 4)^2 + (y - 1)^2 = 2 \][/tex]
Here, the radius [tex]\(r_1 = \sqrt{2}\)[/tex].
### Equation 2:
[tex]\[ x^2 + y^2 + 6x - 4y - 20 = 0 \][/tex]
1. Move the constant term to the right side:
[tex]\[ x^2 + y^2 + 6x - 4y = 20 \][/tex]
2. Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 + 6x + 9 + y^2 - 4y + 4 = 20 + 9 + 4 \][/tex]
3. Factor the perfect squares and solve for the right side:
[tex]\[ (x + 3)^2 + (y - 2)^2 = 33 \][/tex]
Here, the radius [tex]\(r_2 = \sqrt{33}\)[/tex].
### Equation 3:
[tex]\[ 4x^2 + 4y^2 - 16x - 24y + 51 = 0 \][/tex]
1. Divide by 4 to simplify:
[tex]\[ x^2 + y^2 - 4x - 6y + \frac{51}{4} = 0 \][/tex]
2. Move the constant term to the right side:
[tex]\[ x^2 + y^2 - 4x - 6y = -\frac{51}{4} \][/tex]
3. Complete the square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 - 4x + 4 + y^2 - 6y + 9 = -\frac{51}{4} + 4 + 9 \][/tex]
4. Calculate the right side:
[tex]\[ -\frac{51}{4} + 4 + 9 = -\frac{51}{4} + \frac{16}{4} + \frac{36}{4} = -\frac{51 - 16 - 36}{4} = \frac{1}{4} \][/tex]
5. Factor the perfect squares:
[tex]\[ (x - 2)^2 + (y - 3)^2 = \frac{1}{4} \][/tex]
Here, the radius [tex]\(r_3 = \sqrt{\frac{1}{4}} = \frac{1}{2}\)[/tex].
### Summary:
1. Radius for Equation 1: [tex]\( r_1 = \sqrt{2} \)[/tex]
2. Radius for Equation 2: [tex]\( r_2 = \sqrt{33} \)[/tex]
3. Radius for Equation 3: [tex]\( r_3 = \frac{1}{2} \)[/tex]
Smallest radius: [tex]\( \frac{1}{2} \)[/tex] (Equation 3: [tex]\( 4x^2 + 4y^2 - 16x - 24y + 51 = 0 \)[/tex])
Largest radius: [tex]\( \sqrt{33} \)[/tex] (Equation 2: [tex]\( x^2 + y^2 + 6x - 4y - 20 = 0 \)[/tex])
The table should be completed as follows:
| Radius | Equation |
|---------------|------------------------------------------------|
| Smallest | [tex]\( 4 x^2 + 4 y^2 - 16 x - 24 y + 51 = 0 \)[/tex] |
| Largest | [tex]\( x^2 + y^2 + 6 x - 4 y - 20 = 0 \)[/tex] |
