Answer :
Let's analyze the equation of the circle and verify the truth of each statement given.
Step 1: Rewrite the circle equation in standard form
The given circle equation is:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
To rewrite this equation in standard form [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], we need to complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
Step 2: Completing the square
Consider the terms involving [tex]\(x\)[/tex]:
[tex]\[ x^2 - 2x \][/tex]
Complete the square:
[tex]\[ x^2 - 2x = (x - 1)^2 - 1 \][/tex]
Now, consider the terms involving [tex]\(y\)[/tex]:
[tex]\[ y^2 \][/tex]
This term is already a perfect square:
[tex]\[ y^2 = (y - 0)^2 \][/tex]
Substitute these back into the equation:
[tex]\[ (x - 1)^2 - 1 + y^2 - 8 = 0 \][/tex]
Combine constants on one side:
[tex]\[ (x - 1)^2 + (y^2) = 9 \][/tex]
So, the standard form of the circle's equation is:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
Step 3: Identifying the center and radius
From the standard form equation:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
The center [tex]\((h, k)\)[/tex] of the circle is:
[tex]\[ (h, k) = (1, 0) \][/tex]
The radius [tex]\(r\)[/tex] is:
[tex]\[ r = \sqrt{9} = 3 \][/tex]
Step 4: Verifying the statements
1. The radius of the circle is 3 units.
- True. We found the radius to be 3 units.
2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- True. The center [tex]\((1, 0)\)[/tex] means [tex]\(k = 0\)[/tex], so it lies on the [tex]\(x\)[/tex]-axis.
3. The center of the circle lies on the [tex]\(y\)[/tex]-axis.
- False. The center is [tex]\((1, 0)\)[/tex], so it does not lie 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 radius of the circle with the equation [tex]\(x^2 + y^2 = 9\)[/tex] is 3, which matches our circle's radius.
Conclusion
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: Rewrite the circle equation in standard form
The given circle equation is:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
To rewrite this equation in standard form [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], we need to complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
Step 2: Completing the square
Consider the terms involving [tex]\(x\)[/tex]:
[tex]\[ x^2 - 2x \][/tex]
Complete the square:
[tex]\[ x^2 - 2x = (x - 1)^2 - 1 \][/tex]
Now, consider the terms involving [tex]\(y\)[/tex]:
[tex]\[ y^2 \][/tex]
This term is already a perfect square:
[tex]\[ y^2 = (y - 0)^2 \][/tex]
Substitute these back into the equation:
[tex]\[ (x - 1)^2 - 1 + y^2 - 8 = 0 \][/tex]
Combine constants on one side:
[tex]\[ (x - 1)^2 + (y^2) = 9 \][/tex]
So, the standard form of the circle's equation is:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
Step 3: Identifying the center and radius
From the standard form equation:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
The center [tex]\((h, k)\)[/tex] of the circle is:
[tex]\[ (h, k) = (1, 0) \][/tex]
The radius [tex]\(r\)[/tex] is:
[tex]\[ r = \sqrt{9} = 3 \][/tex]
Step 4: Verifying the statements
1. The radius of the circle is 3 units.
- True. We found the radius to be 3 units.
2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- True. The center [tex]\((1, 0)\)[/tex] means [tex]\(k = 0\)[/tex], so it lies on the [tex]\(x\)[/tex]-axis.
3. The center of the circle lies on the [tex]\(y\)[/tex]-axis.
- False. The center is [tex]\((1, 0)\)[/tex], so it does not lie 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 radius of the circle with the equation [tex]\(x^2 + y^2 = 9\)[/tex] is 3, which matches our circle's radius.
Conclusion
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].
