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Sagot :
### Part A: Completing the Square
We start with the given equation:
[tex]\[ x^2 - 4x + y^2 + 8y = -4 \][/tex]
#### Step 1: Completing the square for the [tex]\(x\)[/tex]-terms
Consider the [tex]\(x\)[/tex]-terms [tex]\(x^2 - 4x\)[/tex].
To complete the square:
1. Take half the coefficient of [tex]\(x\)[/tex], which is [tex]\(-4/2 = -2\)[/tex].
2. Square this result, yielding [tex]\((-2)^2 = 4\)[/tex].
3. Add and subtract this square inside the expression to maintain equality.
Thus,
[tex]\[ x^2 - 4x = (x^2 - 4x + 4) - 4 = (x - 2)^2 - 4 \][/tex]
#### Step 2: Completing the square for the [tex]\(y\)[/tex]-terms
Consider the [tex]\(y\)[/tex]-terms [tex]\(y^2 + 8y\)[/tex].
To complete the square:
1. Take half the coefficient of [tex]\(y\)[/tex], which is [tex]\(8/2 = 4\)[/tex].
2. Square this result, yielding [tex]\(4^2 = 16\)[/tex].
3. Add and subtract this square inside the expression to maintain equality.
Thus,
[tex]\[ y^2 + 8y = (y^2 + 8y + 16) - 16 = (y + 4)^2 - 16 \][/tex]
#### Step 3: Rewriting the entire equation
Substitute the completed square forms back into the original equation:
[tex]\[ (x - 2)^2 - 4 + (y + 4)^2 - 16 = -4 \][/tex]
Combine the constants:
[tex]\[ (x - 2)^2 + (y + 4)^2 - 20 = -4 \][/tex]
Isolate the squared terms:
[tex]\[ (x - 2)^2 + (y + 4)^2 = -4 + 20 \][/tex]
[tex]\[ (x - 2)^2 + (y + 4)^2 = 16 \][/tex]
Now, the equation is in standard form:
[tex]\[ (x - 2)^2 + (y + 4)^2 = 16 \][/tex]
### Part B: Identifying the Center and Radius
The standard form of the equation of a circle is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle, and [tex]\(r\)[/tex] is the radius.
#### Identifying the Center
By comparison with [tex]\((x - 2)^2 + (y + 4)^2 = 16\)[/tex], we can see:
[tex]\[ h = 2 \][/tex]
[tex]\[ k = -4 \][/tex]
Thus, the center of the circle is [tex]\((2, -4)\)[/tex].
#### Calculating the Radius
The radius [tex]\(r\)[/tex] is found by taking the square root of the constant on the right-hand side:
[tex]\[ r^2 = 16 \][/tex]
[tex]\[ r = \sqrt{16} = 4 \][/tex]
### Summary
- Center of the circle: [tex]\((2, -4)\)[/tex]
- Radius of the circle: [tex]\(4\)[/tex]
Thus, we have successfully completed the square and identified the center and radius of the circle based on the given equation.
We start with the given equation:
[tex]\[ x^2 - 4x + y^2 + 8y = -4 \][/tex]
#### Step 1: Completing the square for the [tex]\(x\)[/tex]-terms
Consider the [tex]\(x\)[/tex]-terms [tex]\(x^2 - 4x\)[/tex].
To complete the square:
1. Take half the coefficient of [tex]\(x\)[/tex], which is [tex]\(-4/2 = -2\)[/tex].
2. Square this result, yielding [tex]\((-2)^2 = 4\)[/tex].
3. Add and subtract this square inside the expression to maintain equality.
Thus,
[tex]\[ x^2 - 4x = (x^2 - 4x + 4) - 4 = (x - 2)^2 - 4 \][/tex]
#### Step 2: Completing the square for the [tex]\(y\)[/tex]-terms
Consider the [tex]\(y\)[/tex]-terms [tex]\(y^2 + 8y\)[/tex].
To complete the square:
1. Take half the coefficient of [tex]\(y\)[/tex], which is [tex]\(8/2 = 4\)[/tex].
2. Square this result, yielding [tex]\(4^2 = 16\)[/tex].
3. Add and subtract this square inside the expression to maintain equality.
Thus,
[tex]\[ y^2 + 8y = (y^2 + 8y + 16) - 16 = (y + 4)^2 - 16 \][/tex]
#### Step 3: Rewriting the entire equation
Substitute the completed square forms back into the original equation:
[tex]\[ (x - 2)^2 - 4 + (y + 4)^2 - 16 = -4 \][/tex]
Combine the constants:
[tex]\[ (x - 2)^2 + (y + 4)^2 - 20 = -4 \][/tex]
Isolate the squared terms:
[tex]\[ (x - 2)^2 + (y + 4)^2 = -4 + 20 \][/tex]
[tex]\[ (x - 2)^2 + (y + 4)^2 = 16 \][/tex]
Now, the equation is in standard form:
[tex]\[ (x - 2)^2 + (y + 4)^2 = 16 \][/tex]
### Part B: Identifying the Center and Radius
The standard form of the equation of a circle is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle, and [tex]\(r\)[/tex] is the radius.
#### Identifying the Center
By comparison with [tex]\((x - 2)^2 + (y + 4)^2 = 16\)[/tex], we can see:
[tex]\[ h = 2 \][/tex]
[tex]\[ k = -4 \][/tex]
Thus, the center of the circle is [tex]\((2, -4)\)[/tex].
#### Calculating the Radius
The radius [tex]\(r\)[/tex] is found by taking the square root of the constant on the right-hand side:
[tex]\[ r^2 = 16 \][/tex]
[tex]\[ r = \sqrt{16} = 4 \][/tex]
### Summary
- Center of the circle: [tex]\((2, -4)\)[/tex]
- Radius of the circle: [tex]\(4\)[/tex]
Thus, we have successfully completed the square and identified the center and radius of the circle based on the given equation.
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