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Drag each equation to the correct location in the table. Not all equations will be used.

Place the equations that represent circles with the smallest and the largest radius into the table.

Equations:
[tex]2x^2 + 2y^2 + 16x - 4y + 30 = 0[/tex]
[tex]x^2 + y^2 + 6x - 4y - 20 = 0[/tex]
[tex]4x^2 + 4y^2 - 16x - 24y + 51 = 0[/tex]

Sagot :

GinnyAnswer
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] |
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