Sure, let's match each circle equation in its general form to its corresponding equation in the standard form. We'll need to pair the equations correctly.
General Form Equations:
1. [tex]\( x^2 + y^2 - 4x + 12y - 20 = 0 \)[/tex]
2. [tex]\( x^2 + y^2 + 6x - 8y - 10 = 0 \)[/tex]
3. [tex]\( 3x^2 + 3y^2 + 12x + 18y - 15 = 0 \)[/tex]
4. [tex]\( 5x^2 + 5y^2 - 10x + 20y - 30 = 0 \)[/tex]
Standard Form Equations:
1. [tex]\( (x - 6)^2 + (y - 4)^2 = 56 \)[/tex]
2. [tex]\( (x - 2)^2 + (y + 6)^2 = 60 \)[/tex]
3. [tex]\( (x + 2)^2 + (y + 3)^2 = 18 \)[/tex]
4. [tex]\( (x + 1)^2 + (y - 6)^2 = 46 \)[/tex]
Here's the correct matching:
1. [tex]\( x^2 + y^2 - 4x + 12y - 20 = 0 \)[/tex]
- Corresponds to: [tex]\( (x - 6)^2 + (y - 4)^2 = 56 \)[/tex]
2. [tex]\( x^2 + y^2 + 6x - 8y - 10 = 0 \)[/tex]
- Corresponds to: [tex]\( (x - 2)^2 + (y + 6)^2 = 60 \)[/tex]
3. [tex]\( 3x^2 + 3y^2 + 12x + 18y - 15 = 0 \)[/tex]
- Corresponds to: [tex]\( (x + 2)^2 + (y + 3)^2 = 18 \)[/tex]
4. [tex]\( 5x^2 + 5y^2 - 10x + 20y - 30 = 0 \)[/tex]
- Corresponds to: [tex]\( (x + 1)^2 + (y - 6)^2 = 46 \)[/tex]
Our final pairs are:
1. [tex]\( x^2 + y^2 - 4x + 12y - 20 = 0 \)[/tex] → [tex]\( (x - 6)^2 + (y - 4)^2 = 56 \)[/tex]
2. [tex]\( x^2 + y^2 + 6x - 8y - 10 = 0 \)[/tex] → [tex]\( (x - 2)^2 + (y + 6)^2 = 60 \)[/tex]
3. [tex]\( 3x^2 + 3y^2 + 12x + 18y - 15 = 0 \)[/tex] → [tex]\( (x + 2)^2 + (y + 3)^2 = 18 \)[/tex]
4. [tex]\( 5x^2 + 5y^2 - 10x + 20y - 30 = 0 \)[/tex] → [tex]\( (x + 1)^2 + (y - 6)^2 = 46 \)[/tex]
