To determine the properties of the circle from the equation [tex]\( x^2 + y^2 - 2x - 8 = 0 \)[/tex] and to identify the true statements, we need to convert this equation into the standard form for a circle's equation [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]. Here is a step-by-step process:
1. Starting with the given equation:
[tex]\[ x^2 + y^2 - 2x - 8 = 0 \][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms:
[tex]\[ x^2 - 2x \][/tex]
To complete the square, take half of the coefficient of [tex]\(x\)[/tex] (which is -2), square it, and add and subtract this square inside the equation.
[tex]\[ x^2 - 2x \Rightarrow (x - 1)^2 - 1 \][/tex]
3. Rewrite the equation incorporating the completed square:
[tex]\[ (x - 1)^2 - 1 + y^2 - 8 = 0 \][/tex]
4. Combine constants:
[tex]\[ (x - 1)^2 + y^2 - 9 = 0 \][/tex]
[tex]\[ (x - 1)^2 + y^2 = 9 \][/tex]
Now, the equation is in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where the center [tex]\((h, k)\)[/tex] is [tex]\((1, 0)\)[/tex] and the radius [tex]\(r\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex].
Based on this standard form, we can analyze the given statements:
1. The radius of the circle is 3 units.
- True. From the standard form [tex]\((x - 1)^2 + y^2 = 9\)[/tex], the radius [tex]\(r\)[/tex] is [tex]\(\sqrt{9} = 3\)[/tex] units.
2. The center of the circle lies on the [tex]\(x\)[/tex]-axis.
- True. The center [tex]\((1, 0)\)[/tex] has a [tex]\(y\)[/tex]-coordinate of 0, which means it lies on the [tex]\(x\)[/tex]-axis.
3. The center of the circle lies on the [tex]\(y\)[/tex]-axis.
- False. The center [tex]\((1, 0)\)[/tex] has an [tex]\(x\)[/tex]-coordinate of 1, 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 of the equation 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] represents a circle with radius [tex]\(\sqrt{9} = 3\)[/tex], which matches the radius of our circle.
Thus, 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.
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].
