Certainly! To match the equations in general form with their corresponding equations in standard form, follow these steps:
1. General Form: [tex]\(x^2 + y^2 - 4x + 12y - 20 = 0\)[/tex]
- Corresponding Standard Form: [tex]\((x-6)^2 + (y-4)^2 = 56\)[/tex]
2. General Form: [tex]\(x^2 + y^2 + 6x - 8y - 10 = 0\)[/tex]
- Corresponding Standard Form: [tex]\((x-2)^2 + (y+6)^2 = 60\)[/tex]
3. General Form: [tex]\(3x^2 + 3y^2 + 12x + 18y - 15 = 0\)[/tex]
- Corresponding Standard Form: [tex]\((x+2)^2 + (y+3)^2 = 18\)[/tex]
4. General Form: [tex]\(5x^2 + 5y^2 - 10x + 20y - 30 = 0\)[/tex]
- Corresponding Standard Form: [tex]\((x+1)^2 + (y-6)^2 = 46\)[/tex]
The correct pairs are as follows:
[tex]\[
\begin{array}{l|l}
x^2 + y^2 - 4x + 12y - 20 = 0 & (x-6)^2 + (y-4)^2 = 56 \\
x^2 + y^2 + 6x - 8y - 10 = 0 & (x-2)^2 + (y+6)^2 = 60 \\
3x^2 + 3y^2 + 12x + 18y - 15 = 0 & (x+2)^2 + (y+3)^2 = 18 \\
5x^2 + 5y^2 - 10x + 20y - 30 = 0 & (x+1)^2 + (y-6)^2 = 46 \\
\end{array}
\][/tex]
Remember, the equation [tex]\(2x^2 + 2y^2 - 24x - 16y - 8 = 0\)[/tex] did not match with any of the provided standard form equations.
