Let's solve this step-by-step to determine which statements about the given circle equation [tex]$x^2+y^2-2x-8=0$[/tex] are true.
### Step 1: Rewrite the equation in standard form
The given equation of the circle is:
[tex]\[ x^2 + y^2 - 2x - 8 = 0. \][/tex]
We aim to write this equation in the standard form of a circle's equation:
[tex]\[ (x-h)^2 + (y-k)^2 = r^2. \][/tex]
#### Step 1.1: Complete the square for the [tex]$x$[/tex] terms
First, let's focus on the [tex]$x$[/tex] terms:
[tex]\[ x^2 - 2x. \][/tex]
To complete the square, we add and subtract [tex]$\left(\frac{2}{2}\right)^2 = 1$[/tex]:
[tex]\[ x^2 - 2x + 1 - 1 = (x-1)^2 - 1. \][/tex]
#### Step 1.2: Rewrite the equation
Now substitute [tex]$(x-1)^2 - 1$[/tex] back into the original equation:
[tex]\[ (x-1)^2 - 1 + y^2 - 8 = 0. \][/tex]
Combine the constants:
[tex]\[ (x-1)^2 + y^2 - 9 = 0. \][/tex]
[tex]\[ (x-1)^2 + y^2 = 9. \][/tex]
### Step 2: Identify the center and radius
The standard form of the circle's equation is now:
[tex]\[ (x-1)^2 + y^2 = 9. \][/tex]
From the standard form, we can directly read the center and radius:
- The center \((h, k)\) is \((1, 0)\).
- The radius \(r\) is \(\sqrt{9} = 3\).
### Step 3: Verify the true statements
Based on the information we have obtained, we can examine the given options:
1. The radius of the circle is 3 units.
- This is true. We have determined the radius to be 3 units.
2. The center of the circle lies on the [tex]$x$[/tex]-axis.
- This is true. The center is \((1, 0)\), which lies on the [tex]$x$[/tex]-axis.
3. The center of the circle lies on the [tex]$y$[/tex]-axis.
- This is false. The center is located at \((1, 0)\), which is not on the [tex]$y$[/tex]-axis.
4. The standard form of the equation is \((x-1)^2 + y^2 = 3\).
- This is false. The correct standard form is \((x-1)^2 + y^2 = 9\).
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].
- This is true. The radius of the circle with equation [tex]$x^2 + y^2 = 9$[/tex] is also \(\sqrt{9} = 3\).
### Conclusion
The three 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.
3. The radius of this circle is the same as the radius of the circle whose equation is [tex]$x^2 + y^2 = 9$[/tex].
