Answer:
Explanation:
We can start by completing the square to find the standard form of the equation of the circle:
x^2 + y^2 - 2x - 8 = 0
x^2 - 2x + y^2 = 8
(x - 1)^2 + y^2 = 9
(x - 1)^2 + y^2 = (sqrt(3))^2
From this standard form, we can see that the center of the circle is at (1, 0), which lies on the x-axis. Therefore, the statement "The center of the circle lies on the x-axis" is true.
The radius of the circle is the square root of the constant term in the standard form, which is 3. Therefore, the statement "The radius of the circle is 3 units" is false, as the radius is actually sqrt(3) units.
Since the center of the circle lies on the x-axis, it does not lie on the y-axis. Therefore, the statement "The center of the circle lies on the y-axis" is false.
We have already shown that the standard form of the equation of the circle is (x - 1)^2 + y^2 = 3. Therefore, the statement "The standard form of the equation is (x – 1)² + y² = 3" is true.
Finally, the equation x^2 + y^2 = 9 represents a circle of radius 3 centered at the origin, which is not the same as the circle given by the equation x^2 + y^2 - 2x - 8 = 0. Therefore, the statement "The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9" is false.
In summary, the three true statements are:
The center of the circle lies on the x-axis.
The standard form of the equation is (x – 1)² + y² = 3.
The radius of the circle is sqrt(3) units.