Sure, let's carefully match each general form equation with its corresponding standard form equation step-by-step. Given the equations and pairs, we can identify the matches as follows:
1. General Form: [tex]\( x^2 + y^2 - 4x + 12y - 20 = 0 \)[/tex]
- Standard Form: [tex]\( (x - 2)^2 + (y + 6)^2 = 60 \)[/tex]
2. General Form: [tex]\( x^2 + y^2 + 6x - 8y - 10 = 0 \)[/tex]
- Standard Form: [tex]\( (x - 1)^2 + (y - 6)^2 = 46 \)[/tex]
3. General Form: [tex]\( 3x^2 + 3y^2 + 12x + 18y - 15 = 0 \)[/tex]
- Standard Form: [tex]\( (x + 2)^2 + (y + 3)^2 = 18 \)[/tex]
4. General Form: [tex]\( 5x^2 + 5y^2 - 10x + 20y - 30 = 0 \)[/tex]
- Standard Form: [tex]\( x^2 + y^2 + 2x - 12y - 9 = 0 \)[/tex]
5. General Form: [tex]\( 2x^2 + 2y^2 - 24x - 16y - 8 = 0 \)[/tex]
- Standard Form: [tex]\( (x - 6)^2 + (y - 4)^2 = 56 \)[/tex]
So, the matching pairs are:
[tex]\[
\begin{aligned}
&x^2 + y^2 - 4x + 12y - 20 = 0 &\rightarrow (x - 2)^2 + (y + 6)^2 = 60 \\
&x^2 + y^2 + 6x - 8y - 10 = 0 &\rightarrow (x - 1)^2 + (y - 6)^2 = 46 \\
&3x^2 + 3y^2 + 12x + 18y - 15 = 0 &\rightarrow (x + 2)^2 + (y + 3)^2 = 18 \\
&5x^2 + 5y^2 - 10x + 20y - 30 = 0 &\rightarrow x^2 + y^2 + 2x - 12y - 9 = 0 \\
&2x^2 + 2y^2 - 24x - 16y - 8 = 0 &\rightarrow (x - 6)^2 + (y - 4)^2 = 56 \\
\end{aligned}
\][/tex]
