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Sagot :
To determine which statements about the given circle equation [tex]\( x^2 + y^2 - 2x - 8 = 0 \)[/tex] are true, let's carefully analyze the equation step-by-step.
### Step 1: Rewrite the given equation in standard form.
The general form of a circle's equation in standard form is:
[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.
Given equation:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
First, we need to complete the square for the [tex]\(x\)[/tex] terms.
1. Isolate the [tex]\(x\)[/tex] terms and the constants:
[tex]\[ x^2 - 2x + y^2 = 8 \][/tex]
2. Complete the square for [tex]\(x\)[/tex]:
[tex]\[ x^2 - 2x \][/tex]
To complete the square, we take half of the coefficient of [tex]\(x\)[/tex] (which is -2), square it, and add & subtract inside the equation:
[tex]\[ x^2 - 2x + 1 - 1 + y^2 = 8 \][/tex]
[tex]\[ (x - 1)^2 - 1 + y^2 = 8 \][/tex]
3. Move the constants to one side:
[tex]\[ (x - 1)^2 + y^2 - 1 = 8 \][/tex]
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
### Step 2: Identify the center and radius from the standard form.
The equation [tex]\( (x - 1)^2 + y^2 = 9 \)[/tex] reveals that:
- The center [tex]\((h, k)\)[/tex] of the circle is at [tex]\((1, 0)\)[/tex]
- The radius [tex]\(r\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex] units
### Step 3: Evaluate each of the given statements.
1. The radius of the circle is 3 units.
- True. From the equation in standard form, the radius is [tex]\(3\)[/tex].
2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- True. The [tex]\(y\)[/tex]-coordinate of the center is [tex]\(0\)[/tex], placing the center 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 [tex]\(1\)[/tex], not on the [tex]\(y\)[/tex]-axis.
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 equation [tex]\(x^2 + y^2 = 9\)[/tex] also describes a circle with radius [tex]\( \sqrt{9} = 3 \)[/tex].
### Conclusion:
The true statements are:
1. The radius of the circle is 3 units.
2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
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].
Thus, the correct answer is:
[tex]\[ \boxed{[1, 2, 5]} \][/tex]
### Step 1: Rewrite the given equation in standard form.
The general form of a circle's equation in standard form is:
[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.
Given equation:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
First, we need to complete the square for the [tex]\(x\)[/tex] terms.
1. Isolate the [tex]\(x\)[/tex] terms and the constants:
[tex]\[ x^2 - 2x + y^2 = 8 \][/tex]
2. Complete the square for [tex]\(x\)[/tex]:
[tex]\[ x^2 - 2x \][/tex]
To complete the square, we take half of the coefficient of [tex]\(x\)[/tex] (which is -2), square it, and add & subtract inside the equation:
[tex]\[ x^2 - 2x + 1 - 1 + y^2 = 8 \][/tex]
[tex]\[ (x - 1)^2 - 1 + y^2 = 8 \][/tex]
3. Move the constants to one side:
[tex]\[ (x - 1)^2 + y^2 - 1 = 8 \][/tex]
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
### Step 2: Identify the center and radius from the standard form.
The equation [tex]\( (x - 1)^2 + y^2 = 9 \)[/tex] reveals that:
- The center [tex]\((h, k)\)[/tex] of the circle is at [tex]\((1, 0)\)[/tex]
- The radius [tex]\(r\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex] units
### Step 3: Evaluate each of the given statements.
1. The radius of the circle is 3 units.
- True. From the equation in standard form, the radius is [tex]\(3\)[/tex].
2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- True. The [tex]\(y\)[/tex]-coordinate of the center is [tex]\(0\)[/tex], placing the center 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 [tex]\(1\)[/tex], not on the [tex]\(y\)[/tex]-axis.
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 equation [tex]\(x^2 + y^2 = 9\)[/tex] also describes a circle with radius [tex]\( \sqrt{9} = 3 \)[/tex].
### Conclusion:
The true statements are:
1. The radius of the circle is 3 units.
2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
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].
Thus, the correct answer is:
[tex]\[ \boxed{[1, 2, 5]} \][/tex]
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