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Sagot :
To analyze the given equation of the circle, we start by rewriting it in the standard form of a circle's equation: [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex].
The given equation is:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
### Step 1: Completing the Square for [tex]\(x\)[/tex]
First, we focus on the terms involving [tex]\(x\)[/tex]:
[tex]\[ x^2 - 2x \][/tex]
To complete the square, we add and subtract the same value within the equation:
[tex]\[ x^2 - 2x = (x - 1)^2 - 1 \][/tex]
So, the expression can be rewritten as:
[tex]\[ (x - 1)^2 - 1 \][/tex]
### Step 2: Substituting Back into the Original Equation
We substitute [tex]\((x - 1)^2 - 1\)[/tex] back into the equation, replacing [tex]\(x^2 - 2x\)[/tex]:
[tex]\[ (x - 1)^2 - 1 + y^2 - 8 = 0 \][/tex]
### Step 3: Simplifying the Equation
To get the standard form of the circle's equation, we simplify the constant terms:
[tex]\[ (x - 1)^2 + y^2 - 9 = 0 \][/tex]
By adding 9 to both sides, we obtain:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
### Interpretation of the Standard Form
From the standard form [tex]\((x - 1)^2 + y^2 = 9\)[/tex], we can extract the following information:
1. Center of the Circle: The center [tex]\((h, k)\)[/tex] is [tex]\((1, 0)\)[/tex]. So, the x-coordinate of the center is 1 and the y-coordinate is 0. This means:
- The center of the circle lies on the [tex]\(x\)[/tex]-axis.
2. Radius of the Circle: The radius [tex]\(r\)[/tex] is the square root of 9:
[tex]\[ r = \sqrt{9} = 3 \][/tex]
### Evaluating the Statements
Now, we evaluate the given statements:
1. The radius of the circle is 3 units: This statement is True.
2. The center of the circle lies on the x-axis: Since the center is [tex]\((1, 0)\)[/tex], it does lie on the [tex]\(x\)[/tex]-axis. This statement is True.
3. The center of the circle lies on the y-axis: The center does not lie on the [tex]\(y\)[/tex]-axis since the x-coordinate is not zero. This statement is False.
4. The standard form of the equation is [tex]\((x-1)^2 + y^2 = 3\)[/tex]: The correct standard form of the equation is [tex]\((x-1)^2 + y^2 = 9\)[/tex]. This statement is False.
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]: The radius of the circle given by [tex]\(x^2 + y^2 = 9\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex], which is the same as the radius of our circle. This statement is True.
### Conclusion
The three correct statements are:
1. The radius of the circle is 3 units.
2. The center of the circle lies on the x-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].
So, the final results are:
- True, True, True, 1, 0, 3, '(x - 1)^2 + y^2 = 9'
The given equation is:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
### Step 1: Completing the Square for [tex]\(x\)[/tex]
First, we focus on the terms involving [tex]\(x\)[/tex]:
[tex]\[ x^2 - 2x \][/tex]
To complete the square, we add and subtract the same value within the equation:
[tex]\[ x^2 - 2x = (x - 1)^2 - 1 \][/tex]
So, the expression can be rewritten as:
[tex]\[ (x - 1)^2 - 1 \][/tex]
### Step 2: Substituting Back into the Original Equation
We substitute [tex]\((x - 1)^2 - 1\)[/tex] back into the equation, replacing [tex]\(x^2 - 2x\)[/tex]:
[tex]\[ (x - 1)^2 - 1 + y^2 - 8 = 0 \][/tex]
### Step 3: Simplifying the Equation
To get the standard form of the circle's equation, we simplify the constant terms:
[tex]\[ (x - 1)^2 + y^2 - 9 = 0 \][/tex]
By adding 9 to both sides, we obtain:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
### Interpretation of the Standard Form
From the standard form [tex]\((x - 1)^2 + y^2 = 9\)[/tex], we can extract the following information:
1. Center of the Circle: The center [tex]\((h, k)\)[/tex] is [tex]\((1, 0)\)[/tex]. So, the x-coordinate of the center is 1 and the y-coordinate is 0. This means:
- The center of the circle lies on the [tex]\(x\)[/tex]-axis.
2. Radius of the Circle: The radius [tex]\(r\)[/tex] is the square root of 9:
[tex]\[ r = \sqrt{9} = 3 \][/tex]
### Evaluating the Statements
Now, we evaluate the given statements:
1. The radius of the circle is 3 units: This statement is True.
2. The center of the circle lies on the x-axis: Since the center is [tex]\((1, 0)\)[/tex], it does lie on the [tex]\(x\)[/tex]-axis. This statement is True.
3. The center of the circle lies on the y-axis: The center does not lie on the [tex]\(y\)[/tex]-axis since the x-coordinate is not zero. This statement is False.
4. The standard form of the equation is [tex]\((x-1)^2 + y^2 = 3\)[/tex]: The correct standard form of the equation is [tex]\((x-1)^2 + y^2 = 9\)[/tex]. This statement is False.
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]: The radius of the circle given by [tex]\(x^2 + y^2 = 9\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex], which is the same as the radius of our circle. This statement is True.
### Conclusion
The three correct statements are:
1. The radius of the circle is 3 units.
2. The center of the circle lies on the x-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].
So, the final results are:
- True, True, True, 1, 0, 3, '(x - 1)^2 + y^2 = 9'
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