Answer :
To determine which statements are true regarding the given circle's equation [tex]\( x^2 + y^2 - 2x - 8 = 0 \)[/tex], we need to rewrite the equation in standard form and extract information about the circle's center and radius.
### Step-by-Step Process
1. Rewrite the equation in standard form:
The general form for a circle's equation 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.
2. Complete the square for the [tex]\(x\)[/tex]-terms:
The given equation is:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
First, focus on the [tex]\(x\)[/tex]-terms:
[tex]\[ x^2 - 2x \][/tex]
To complete the square, add and subtract 1 inside the equation:
[tex]\[ x^2 - 2x + 1 - 1 = (x - 1)^2 - 1 \][/tex]
Now, substitute this back into the original equation:
[tex]\[ (x - 1)^2 - 1 + y^2 - 8 = 0 \][/tex]
3. Combine and simplify the equation:
[tex]\[ (x - 1)^2 + y^2 - 9 = 0 \][/tex]
Adding 9 to both sides to move the constant term:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
This is now in the standard form [tex]\( (x - h)^2 + (y - k)^2 = r^2 \)[/tex].
4. Identify the center and radius:
From the standard form [tex]\( (x - 1)^2 + y^2 = 9 \)[/tex], we see:
- The center [tex]\((h, k)\)[/tex] is [tex]\((1, 0)\)[/tex].
- The radius [tex]\( r \)[/tex] is [tex]\(\sqrt{9} = 3 \)[/tex].
### Verification of Statements:
1. The radius of the circle is 3 units.
- True. The radius [tex]\( r = 3 \)[/tex].
2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- True. The center is [tex]\((1, 0)\)[/tex], which lies on the [tex]\(x\)[/tex]-axis (since [tex]\( y = 0 \)[/tex]).
3. The center of the circle lies on the [tex]\(y\)[/tex]-axis.
- False. The center is [tex]\((1, 0)\)[/tex], which does not lie on the [tex]\(y\)[/tex]-axis (since [tex]\( x \neq 0 \)[/tex]).
4. The standard form of the equation is [tex]\((x - 1)^2 + y^2 = 3\)[/tex].
- False. The 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] has a radius [tex]\( \sqrt{9} = 3 \)[/tex], which is the same as the radius of our circle.
### Statements Summary:
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 selected true statements are:
[tex]\[ 1, 2, 5 \][/tex]
### Step-by-Step Process
1. Rewrite the equation in standard form:
The general form for a circle's equation 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.
2. Complete the square for the [tex]\(x\)[/tex]-terms:
The given equation is:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
First, focus on the [tex]\(x\)[/tex]-terms:
[tex]\[ x^2 - 2x \][/tex]
To complete the square, add and subtract 1 inside the equation:
[tex]\[ x^2 - 2x + 1 - 1 = (x - 1)^2 - 1 \][/tex]
Now, substitute this back into the original equation:
[tex]\[ (x - 1)^2 - 1 + y^2 - 8 = 0 \][/tex]
3. Combine and simplify the equation:
[tex]\[ (x - 1)^2 + y^2 - 9 = 0 \][/tex]
Adding 9 to both sides to move the constant term:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
This is now in the standard form [tex]\( (x - h)^2 + (y - k)^2 = r^2 \)[/tex].
4. Identify the center and radius:
From the standard form [tex]\( (x - 1)^2 + y^2 = 9 \)[/tex], we see:
- The center [tex]\((h, k)\)[/tex] is [tex]\((1, 0)\)[/tex].
- The radius [tex]\( r \)[/tex] is [tex]\(\sqrt{9} = 3 \)[/tex].
### Verification of Statements:
1. The radius of the circle is 3 units.
- True. The radius [tex]\( r = 3 \)[/tex].
2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- True. The center is [tex]\((1, 0)\)[/tex], which lies on the [tex]\(x\)[/tex]-axis (since [tex]\( y = 0 \)[/tex]).
3. The center of the circle lies on the [tex]\(y\)[/tex]-axis.
- False. The center is [tex]\((1, 0)\)[/tex], which does not lie on the [tex]\(y\)[/tex]-axis (since [tex]\( x \neq 0 \)[/tex]).
4. The standard form of the equation is [tex]\((x - 1)^2 + y^2 = 3\)[/tex].
- False. The 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] has a radius [tex]\( \sqrt{9} = 3 \)[/tex], which is the same as the radius of our circle.
### Statements Summary:
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 selected true statements are:
[tex]\[ 1, 2, 5 \][/tex]
