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Sagot :
To match the given equations from the table to the corresponding equations from the result list, let's follow a structured process. We want to transform each general quadratic equation into its corresponding completed square form and match it with given forms.
Let's do one example from the table and match it to the correct equation from the numbered list:
1. Starting with [tex]\(x^2 + y^2 - 4x + 12y - 20\)[/tex]:
Group and complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms separately:
[tex]\[ \text{Rewrite the equation: } x^2 - 4x + y^2 + 12y - 20 = 0 \][/tex]
Complete the square for [tex]\(x\)[/tex]:
[tex]\[ x^2 - 4x \Rightarrow (x - 2)^2 - 4 \][/tex]
Complete the square for [tex]\(y\)[/tex]:
[tex]\[ y^2 + 12y \Rightarrow (y + 6)^2 - 36 \][/tex]
Substitute these back into the equation:
[tex]\[ (x - 2)^2 - 4 + (y + 6)^2 - 36 - 20 = 0 \][/tex]
Combine and simplify:
[tex]\[ (x - 2)^2 + (y + 6)^2 - 60 = 0 \][/tex]
From the given matched result, this corresponds to equation [tex]\((x - 2)^2 + (y + 6)^2 - 60 = 0\)[/tex], which matches [tex]\(eq4\)[/tex].
Now match each equation from the table using the same technique:
2. [tex]\( (x - 6)^2 + (y - 4)^2 = 56 \)[/tex] is directly in its completed square form, corresponding to [tex]\(eq2\)[/tex].
3. [tex]\( x^2 + y^2 + 6x - 8y - 10 \)[/tex]:
Completing the square:
[tex]\[ x^2 + 6x + y^2 - 8y = 10 \][/tex]
[tex]\[ (x + 3)^2 - 9 + (y - 4)^2 - 16 = 10 \][/tex]
[tex]\[ (x + 3)^2 + (y - 4)^2 = 35 \Rightarrow (x + 3)^2 + (y - 4)^2 - 35 \Rightarrow 0 \][/tex]
Which matches [tex]\(eq3\)[/tex], upon simplified matching [tex]\(eq9 \rightarrow 0 \)[/tex].
4. [tex]\( (x + 2)^2 + (y + 3)^2 = 18 \)[/tex] is in its completed square form, corresponding to [tex]\(eq6\)[/tex].
5. [tex]\( 5x^2 + 5y^2 - 10x + 20y - 30 \)[/tex]:
Simplify by dividing all terms by 5:
[tex]\[ x^2 + y^2 - 2x + 4y - 6 = 0 \][/tex]
Completing the square:
[tex]\[ (x - 1)^2 - 1 + (y + 2)^2 - 4 = 6 \][/tex]
[tex]\[ (x - 1)^2 + (y + 2)^2 = 11 \Rightarrow (x - 1)^2 + (y + 2)^2 - 11 = 0 \][/tex]
which corresponds to [tex]\(eq7\)[/tex].
6. [tex]\( (x + 1)^2 + (y - 6)^2 = 46 \)[/tex] is in its completed square form, corresponding to [tex]\(eq8\)[/tex].
7. [tex]\( 2x^2 + 2y^2 - 24x - 16y - 8 \)[/tex]:
Simplify by dividing all terms by 2:
[tex]\[ x^2 + y^2 - 12x - 8y = 4 \][/tex]
Completing the square:
[tex]\[ (x - 6)^2 - 36 + (y - 4)^2 - 16 = 4 \][/tex]
[tex]\[ (x - 6)^2 + (y - 4)^2 = 56 \Rightarrow (x - 6)^2 + (y - 4)^2 - 56 = 0 \][/tex]
corresponding to [tex]\(eq9\)[/tex].
8. [tex]\( x^2 + y^2 + 2x - 12y - 9 \)[/tex]:
Completing the square:
[tex]\[ x^2 + 2x + y^2 - 12y = 9 \][/tex]
[tex]\[ (x + 1)^2 - 1 + (y - 6)^2 - 36 = 9 \][/tex]
[tex]\[ (x + 1)^2 + (y - 6)^2 = 46 \Rightarrow (x + 1)^2 + (y - 6)^2 - 46 = 0 \][/tex]
corresponding to [tex]\(eq10\)[/tex].
___
In summary, all equations from the table with their transformed forms matched as per the details:
[tex]\[ \begin{array}{ll} \text{eq1: } x^2 + y^2 - 4x + 12y - 20 & \text{matches to } \text{eq4: } (x - 2)^2 + (y + 6)^2 - 60 \\ \text{eq2: } (x - 6)^2 + (y - 4)^2 = 56 & \text{matches to } \text{eq2: } (x - 6)^2 + (y - 4)^2 - 56 \\ \text{eq3: } x^2 + y^2 + 6x - 8y - 10 & \text{matches to } \text{eq3: } x^2 + y^2 + 6x - 8y - 10 \\ \text{eq4: } (x - 2)^2 + (y + 6)^2 = 60 & \text{matches to } \text{eq4: } (x - 2)^2 + (y + 6)^2 - 60 \\ \text{eq5: } 3x^2 + 3y^2 + 12x + 18y - 15 & \text{matches to } \text{eq7: } 5x^2 + 5y^2 - 10x + 20y - 30 \\ \text{eq6: } (x + 2)^2 + (y + 3)^2 = 18 & \text{matches to } \text{eq6: } x^2 + y^2 + 6x - 8y - 10 \\ \text{eq7: } 5x^2 + 5y^2 - 10x + 20y - 30 & \text{matches to } \text{eq9: } 2x^2 + 2y^2 - 24x - 16y - 8 \\ \text{eq8: } (x + 1)^2 + (y - 6)^2 = 46 & \text{matches to } \text{eq8: } (x + 1)^2 + (y - 6)^2 - 46 \\ \text{eq9: } 2x^2 + 2y^2 - 24x - 16y - 8 & \text{matches to } \text{eq10: } x^2 + y^2 + 2x - 12y - 9 \\ \text{eq10: } x^2 + y^2 + 2x - 12y - 9 & \text{matches to } \text{eq1: } (x - 6)^2 + (y - 4)^2 - 56 - 20 \end{array} \][/tex]
Let's do one example from the table and match it to the correct equation from the numbered list:
1. Starting with [tex]\(x^2 + y^2 - 4x + 12y - 20\)[/tex]:
Group and complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms separately:
[tex]\[ \text{Rewrite the equation: } x^2 - 4x + y^2 + 12y - 20 = 0 \][/tex]
Complete the square for [tex]\(x\)[/tex]:
[tex]\[ x^2 - 4x \Rightarrow (x - 2)^2 - 4 \][/tex]
Complete the square for [tex]\(y\)[/tex]:
[tex]\[ y^2 + 12y \Rightarrow (y + 6)^2 - 36 \][/tex]
Substitute these back into the equation:
[tex]\[ (x - 2)^2 - 4 + (y + 6)^2 - 36 - 20 = 0 \][/tex]
Combine and simplify:
[tex]\[ (x - 2)^2 + (y + 6)^2 - 60 = 0 \][/tex]
From the given matched result, this corresponds to equation [tex]\((x - 2)^2 + (y + 6)^2 - 60 = 0\)[/tex], which matches [tex]\(eq4\)[/tex].
Now match each equation from the table using the same technique:
2. [tex]\( (x - 6)^2 + (y - 4)^2 = 56 \)[/tex] is directly in its completed square form, corresponding to [tex]\(eq2\)[/tex].
3. [tex]\( x^2 + y^2 + 6x - 8y - 10 \)[/tex]:
Completing the square:
[tex]\[ x^2 + 6x + y^2 - 8y = 10 \][/tex]
[tex]\[ (x + 3)^2 - 9 + (y - 4)^2 - 16 = 10 \][/tex]
[tex]\[ (x + 3)^2 + (y - 4)^2 = 35 \Rightarrow (x + 3)^2 + (y - 4)^2 - 35 \Rightarrow 0 \][/tex]
Which matches [tex]\(eq3\)[/tex], upon simplified matching [tex]\(eq9 \rightarrow 0 \)[/tex].
4. [tex]\( (x + 2)^2 + (y + 3)^2 = 18 \)[/tex] is in its completed square form, corresponding to [tex]\(eq6\)[/tex].
5. [tex]\( 5x^2 + 5y^2 - 10x + 20y - 30 \)[/tex]:
Simplify by dividing all terms by 5:
[tex]\[ x^2 + y^2 - 2x + 4y - 6 = 0 \][/tex]
Completing the square:
[tex]\[ (x - 1)^2 - 1 + (y + 2)^2 - 4 = 6 \][/tex]
[tex]\[ (x - 1)^2 + (y + 2)^2 = 11 \Rightarrow (x - 1)^2 + (y + 2)^2 - 11 = 0 \][/tex]
which corresponds to [tex]\(eq7\)[/tex].
6. [tex]\( (x + 1)^2 + (y - 6)^2 = 46 \)[/tex] is in its completed square form, corresponding to [tex]\(eq8\)[/tex].
7. [tex]\( 2x^2 + 2y^2 - 24x - 16y - 8 \)[/tex]:
Simplify by dividing all terms by 2:
[tex]\[ x^2 + y^2 - 12x - 8y = 4 \][/tex]
Completing the square:
[tex]\[ (x - 6)^2 - 36 + (y - 4)^2 - 16 = 4 \][/tex]
[tex]\[ (x - 6)^2 + (y - 4)^2 = 56 \Rightarrow (x - 6)^2 + (y - 4)^2 - 56 = 0 \][/tex]
corresponding to [tex]\(eq9\)[/tex].
8. [tex]\( x^2 + y^2 + 2x - 12y - 9 \)[/tex]:
Completing the square:
[tex]\[ x^2 + 2x + y^2 - 12y = 9 \][/tex]
[tex]\[ (x + 1)^2 - 1 + (y - 6)^2 - 36 = 9 \][/tex]
[tex]\[ (x + 1)^2 + (y - 6)^2 = 46 \Rightarrow (x + 1)^2 + (y - 6)^2 - 46 = 0 \][/tex]
corresponding to [tex]\(eq10\)[/tex].
___
In summary, all equations from the table with their transformed forms matched as per the details:
[tex]\[ \begin{array}{ll} \text{eq1: } x^2 + y^2 - 4x + 12y - 20 & \text{matches to } \text{eq4: } (x - 2)^2 + (y + 6)^2 - 60 \\ \text{eq2: } (x - 6)^2 + (y - 4)^2 = 56 & \text{matches to } \text{eq2: } (x - 6)^2 + (y - 4)^2 - 56 \\ \text{eq3: } x^2 + y^2 + 6x - 8y - 10 & \text{matches to } \text{eq3: } x^2 + y^2 + 6x - 8y - 10 \\ \text{eq4: } (x - 2)^2 + (y + 6)^2 = 60 & \text{matches to } \text{eq4: } (x - 2)^2 + (y + 6)^2 - 60 \\ \text{eq5: } 3x^2 + 3y^2 + 12x + 18y - 15 & \text{matches to } \text{eq7: } 5x^2 + 5y^2 - 10x + 20y - 30 \\ \text{eq6: } (x + 2)^2 + (y + 3)^2 = 18 & \text{matches to } \text{eq6: } x^2 + y^2 + 6x - 8y - 10 \\ \text{eq7: } 5x^2 + 5y^2 - 10x + 20y - 30 & \text{matches to } \text{eq9: } 2x^2 + 2y^2 - 24x - 16y - 8 \\ \text{eq8: } (x + 1)^2 + (y - 6)^2 = 46 & \text{matches to } \text{eq8: } (x + 1)^2 + (y - 6)^2 - 46 \\ \text{eq9: } 2x^2 + 2y^2 - 24x - 16y - 8 & \text{matches to } \text{eq10: } x^2 + y^2 + 2x - 12y - 9 \\ \text{eq10: } x^2 + y^2 + 2x - 12y - 9 & \text{matches to } \text{eq1: } (x - 6)^2 + (y - 4)^2 - 56 - 20 \end{array} \][/tex]
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