Consider a circle whose equation is [tex]x^2 + y^2 - 2x - 8 = 0[/tex]. Which statements are true? Select three options.

A. The radius of the circle is 3 units.
B. The center of the circle lies on the [tex]x[/tex]-axis.
C. The center of the circle lies on the [tex]y[/tex]-axis.
D. The standard form of the equation is [tex](x - 1)^2 + y^2 = 3[/tex].
E. The radius of this circle is the same as the radius of the circle whose equation is [tex]x^2 + y^2 = 9[/tex].



Answer :

To analyze the circle described by the equation [tex]\( x^2 + y^2 - 2x - 8 = 0 \)[/tex] and determine the truth of the given statements, we first need to transform this equation into its standard form.

### Step 1: Completing the Square

1. Original equation:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]

2. Rearrange to group the [tex]\( x \)[/tex]-terms and [tex]\( y \)[/tex]-terms:
[tex]\[ x^2 - 2x + y^2 - 8 = 0 \][/tex]

3. Move the constant term to the right side of the equation:
[tex]\[ x^2 - 2x + y^2 = 8 \][/tex]

4. Complete the square for the [tex]\( x \)[/tex]-terms:
- Take half the coefficient of [tex]\( x \)[/tex], which is [tex]\(-2\)[/tex], and square it: [tex]\(\left(\frac{-2}{2}\right)^2 = 1\)[/tex].
- Add and subtract this square inside the equation:
[tex]\[ x^2 - 2x + 1 - 1 + y^2 = 8 \][/tex]

Rewrite the equation as:
[tex]\[ (x - 1)^2 - 1 + y^2 = 8 \][/tex]

5. Simplify the equation by moving the [tex]\(-1\)[/tex] to the right side:
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]

### Step 2: Analyzing the Standard Form

Now, the equation is in the standard form [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.

- Center: [tex]\((h, k) = (1, 0)\)[/tex]
- Radius: [tex]\( r = \sqrt{9} = 3 \)[/tex]

### Step 3: Evaluate Each Statement

1. The radius of the circle is 3 units.

- True. We determined that [tex]\( r = 3 \)[/tex].

2. The center of the circle lies on the [tex]\( x \)[/tex]-axis.

- True. The [tex]\( y \)[/tex]-coordinate of the center is 0, so it lies 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 1, not 0.

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 circle [tex]\( x^2 + y^2 = 9 \)[/tex] has radius [tex]\( \sqrt{9} = 3 \)[/tex], which matches our circle's radius.

### Conclusion:

Given the evaluation of the statements, 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].