Question

Consider a circle whose equation is x2 + y2 – 2x – 8 = 0. Which statements are true? Select three options.

The radius of the circle is 3 units.
The center of the circle lies on the x-axis.
The center of the circle lies on the y-axis.
The standard form of the equation is (x – 1)² + y² = 3.
The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

Answer 1

1. The radius of the circle is 3 units.
2. The center of the circle does not lie on the x-axis or the y-axis.
3. The standard form of the equation is (x - 1)² + y² = 3.

Explanation:

To determine the properties of the circle whose equation is x² + y² - 2x - 8 = 0, we can complete the square as follows:

x² - 2x + y² = 8

(x - 1)² + y² = 9

From this, we can see that the center of the circle is at the point (1, 0) and the radius is 3 units. Therefore, the statement "The center of the circle lies on the x-axis" is false and "The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9" is true.

Finally, we can rewrite the equation of the circle in the standard form, which is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. In this case, we have (x - 1)² + y² = 3², which confirms the statement "The standard form of the equation is (x - 1)² + y² = 3".