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Sagot :
To determine which statements are true about the circle given by the equation [tex]\(x^2 + y^2 + 4x - 6y - 36 = 0\)[/tex], let's convert it to the standard form of the equation of a circle.
### Step-by-Step Solution:
1. Starting Equation:
The given equation is:
[tex]\[ x^2 + y^2 + 4x - 6y - 36 = 0 \][/tex]
2. Move the Constant Term to the Right Side:
To complete the square later, isolate the constant term on the right side.
[tex]\[ x^2 + y^2 + 4x -6 y = 36 \][/tex]
3. Complete the Square for the [tex]\(x\)[/tex] Terms:
- For [tex]\(x^2 + 4x\)[/tex], add and subtract 4 (since [tex]\((4/2)^2 = 4\)[/tex]):
[tex]\[ x^2 + 4x = (x + 2)^2 - 4 \][/tex]
4. Complete the Square for the [tex]\(y\)[/tex] Terms:
- For [tex]\(y^2 - 6y\)[/tex], add and subtract 9 (since [tex]\((-6/2)^2 = 9\)[/tex]):
[tex]\[ y^2 - 6y = (y - 3)^2 - 9 \][/tex]
5. Incorporate These into the Equation:
Substitute the completed squares back into the equation:
[tex]\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 = 36 \][/tex]
6. Simplify the Expression:
Combine the constants on the right side:
[tex]\[ (x + 2)^2 + (y - 3)^2 - 13 = 36 \][/tex]
[tex]\[ (x + 2)^2 + (y - 3)^2 = 49 \][/tex]
7. Standard Form of the Circle:
The equation [tex]\((x + 2)^2 + (y - 3)^2 = 49\)[/tex] is in standard form.
8. Identify the Center and Radius:
- The center of the circle [tex]\((h, k)\)[/tex] is [tex]\((-2, 3)\)[/tex].
- The radius [tex]\(r\)[/tex] is given by [tex]\(\sqrt{49} = 7\)[/tex] units.
### Correct Statements:
Considering the choices provided:
1. "To begin converting the equation to standard form, subtract 36 from both sides." - This is incorrect. We actually isolate the constant term.
2. "To complete the square for the [tex]\(x\)[/tex] terms, add 4 to both sides." - Correct, since completing the square for [tex]\(x^2 + 4x\)[/tex] involves adding 4.
3. "The center of the circle is at [tex]\((-2, 3)\)[/tex]." - Correct.
4. "The center of the circle is at [tex]\((4, -6)\)[/tex]." - Incorrect.
5. "The radius of the circle is 6 units." - Incorrect. The radius is not 6 units.
6. "The radius of the circle is 49 units." - Incorrect. The radius is not 49 units; it is [tex]\(\sqrt{49} = 7\)[/tex] units.
### Final Correct Statements:
- "To complete the square for the [tex]\(x\)[/tex] terms, add 4 to both sides."
- "The center of the circle is at [tex]\((-2, 3)\)[/tex]."
These are the statements that are accurate based on the conversion of the given equation to the standard form of a circle.
### Step-by-Step Solution:
1. Starting Equation:
The given equation is:
[tex]\[ x^2 + y^2 + 4x - 6y - 36 = 0 \][/tex]
2. Move the Constant Term to the Right Side:
To complete the square later, isolate the constant term on the right side.
[tex]\[ x^2 + y^2 + 4x -6 y = 36 \][/tex]
3. Complete the Square for the [tex]\(x\)[/tex] Terms:
- For [tex]\(x^2 + 4x\)[/tex], add and subtract 4 (since [tex]\((4/2)^2 = 4\)[/tex]):
[tex]\[ x^2 + 4x = (x + 2)^2 - 4 \][/tex]
4. Complete the Square for the [tex]\(y\)[/tex] Terms:
- For [tex]\(y^2 - 6y\)[/tex], add and subtract 9 (since [tex]\((-6/2)^2 = 9\)[/tex]):
[tex]\[ y^2 - 6y = (y - 3)^2 - 9 \][/tex]
5. Incorporate These into the Equation:
Substitute the completed squares back into the equation:
[tex]\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 = 36 \][/tex]
6. Simplify the Expression:
Combine the constants on the right side:
[tex]\[ (x + 2)^2 + (y - 3)^2 - 13 = 36 \][/tex]
[tex]\[ (x + 2)^2 + (y - 3)^2 = 49 \][/tex]
7. Standard Form of the Circle:
The equation [tex]\((x + 2)^2 + (y - 3)^2 = 49\)[/tex] is in standard form.
8. Identify the Center and Radius:
- The center of the circle [tex]\((h, k)\)[/tex] is [tex]\((-2, 3)\)[/tex].
- The radius [tex]\(r\)[/tex] is given by [tex]\(\sqrt{49} = 7\)[/tex] units.
### Correct Statements:
Considering the choices provided:
1. "To begin converting the equation to standard form, subtract 36 from both sides." - This is incorrect. We actually isolate the constant term.
2. "To complete the square for the [tex]\(x\)[/tex] terms, add 4 to both sides." - Correct, since completing the square for [tex]\(x^2 + 4x\)[/tex] involves adding 4.
3. "The center of the circle is at [tex]\((-2, 3)\)[/tex]." - Correct.
4. "The center of the circle is at [tex]\((4, -6)\)[/tex]." - Incorrect.
5. "The radius of the circle is 6 units." - Incorrect. The radius is not 6 units.
6. "The radius of the circle is 49 units." - Incorrect. The radius is not 49 units; it is [tex]\(\sqrt{49} = 7\)[/tex] units.
### Final Correct Statements:
- "To complete the square for the [tex]\(x\)[/tex] terms, add 4 to both sides."
- "The center of the circle is at [tex]\((-2, 3)\)[/tex]."
These are the statements that are accurate based on the conversion of the given equation to the standard form of a circle.
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