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Sagot :
To analyze the circle described by the equation [tex]\( x^2 + y^2 - 2x - 8 = 0 \)[/tex] and determine the truth of the given statements, we first need to transform this equation into its standard form.
### Step 1: Completing the Square
1. Original equation:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
2. Rearrange to group the [tex]\( x \)[/tex]-terms and [tex]\( y \)[/tex]-terms:
[tex]\[ x^2 - 2x + y^2 - 8 = 0 \][/tex]
3. Move the constant term to the right side of the equation:
[tex]\[ x^2 - 2x + y^2 = 8 \][/tex]
4. Complete the square for the [tex]\( x \)[/tex]-terms:
- Take half the coefficient of [tex]\( x \)[/tex], which is [tex]\(-2\)[/tex], and square it: [tex]\(\left(\frac{-2}{2}\right)^2 = 1\)[/tex].
- Add and subtract this square inside the equation:
[tex]\[ x^2 - 2x + 1 - 1 + y^2 = 8 \][/tex]
Rewrite the equation as:
[tex]\[ (x - 1)^2 - 1 + y^2 = 8 \][/tex]
5. Simplify the equation by moving the [tex]\(-1\)[/tex] to the right side:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
### Step 2: Analyzing the Standard Form
Now, the equation is in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\( r \)[/tex] is the radius.
- Center: [tex]\((h, k) = (1, 0)\)[/tex]
- Radius: [tex]\( r = \sqrt{9} = 3 \)[/tex]
### Step 3: Evaluate Each Statement
1. The radius of the circle is 3 units.
- True. We determined that [tex]\( r = 3 \)[/tex].
2. The center of the circle lies on the [tex]\( x \)[/tex]-axis.
- True. The [tex]\( y \)[/tex]-coordinate of the center is 0, so it lies on the [tex]\( x \)[/tex]-axis.
3. The center of the circle lies on the [tex]\( y \)[/tex]-axis.
- False. The [tex]\( x \)[/tex]-coordinate of the center is 1, not 0.
4. The standard form of the equation is [tex]\( (x - 1)^2 + y^2 = 3 \)[/tex].
- False. The correct standard form is [tex]\( (x - 1)^2 + y^2 = 9 \)[/tex].
5. The radius of this circle is the same as the radius of the circle whose equation is [tex]\( x^2 + y^2 = 9 \)[/tex].
- True. The circle [tex]\( x^2 + y^2 = 9 \)[/tex] has radius [tex]\( \sqrt{9} = 3 \)[/tex], which matches our circle's radius.
### Conclusion:
Given the evaluation of the statements, the true statements are:
- The radius of the circle is 3 units.
- The center of the circle lies on the [tex]\( x \)[/tex]-axis.
- The radius of this circle is the same as the radius of the circle whose equation is [tex]\( x^2 + y^2 = 9 \)[/tex].
### Step 1: Completing the Square
1. Original equation:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
2. Rearrange to group the [tex]\( x \)[/tex]-terms and [tex]\( y \)[/tex]-terms:
[tex]\[ x^2 - 2x + y^2 - 8 = 0 \][/tex]
3. Move the constant term to the right side of the equation:
[tex]\[ x^2 - 2x + y^2 = 8 \][/tex]
4. Complete the square for the [tex]\( x \)[/tex]-terms:
- Take half the coefficient of [tex]\( x \)[/tex], which is [tex]\(-2\)[/tex], and square it: [tex]\(\left(\frac{-2}{2}\right)^2 = 1\)[/tex].
- Add and subtract this square inside the equation:
[tex]\[ x^2 - 2x + 1 - 1 + y^2 = 8 \][/tex]
Rewrite the equation as:
[tex]\[ (x - 1)^2 - 1 + y^2 = 8 \][/tex]
5. Simplify the equation by moving the [tex]\(-1\)[/tex] to the right side:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
### Step 2: Analyzing the Standard Form
Now, the equation is in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\( r \)[/tex] is the radius.
- Center: [tex]\((h, k) = (1, 0)\)[/tex]
- Radius: [tex]\( r = \sqrt{9} = 3 \)[/tex]
### Step 3: Evaluate Each Statement
1. The radius of the circle is 3 units.
- True. We determined that [tex]\( r = 3 \)[/tex].
2. The center of the circle lies on the [tex]\( x \)[/tex]-axis.
- True. The [tex]\( y \)[/tex]-coordinate of the center is 0, so it lies on the [tex]\( x \)[/tex]-axis.
3. The center of the circle lies on the [tex]\( y \)[/tex]-axis.
- False. The [tex]\( x \)[/tex]-coordinate of the center is 1, not 0.
4. The standard form of the equation is [tex]\( (x - 1)^2 + y^2 = 3 \)[/tex].
- False. The correct standard form is [tex]\( (x - 1)^2 + y^2 = 9 \)[/tex].
5. The radius of this circle is the same as the radius of the circle whose equation is [tex]\( x^2 + y^2 = 9 \)[/tex].
- True. The circle [tex]\( x^2 + y^2 = 9 \)[/tex] has radius [tex]\( \sqrt{9} = 3 \)[/tex], which matches our circle's radius.
### Conclusion:
Given the evaluation of the statements, the true statements are:
- The radius of the circle is 3 units.
- The center of the circle lies on the [tex]\( x \)[/tex]-axis.
- The radius of this circle is the same as the radius of the circle whose equation is [tex]\( x^2 + y^2 = 9 \)[/tex].
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