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

Answer: To determine which statements are true, we can start by putting the equation of the circle in standard form:

x^2 + y^2 - 2x - 8 = 0

(x^2 - 2x) + y^2 = 8

Completing the square for the x-terms, we add and subtract (2/2)^2 = 1:

(x^2 - 2x + 1 - 1) + y^2 = 8

(x - 1)^2 + y^2 = 9

This is the equation of a circle with center (1, 0) and radius 3. Therefore, the following statements are true:

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 this circle is 3 units.

The statement "The radius of the circle is 3 units" is true because we just found that the equation is equivalent to (x – 1)² + y² = 9, which has radius 3.

The statement "The center of the circle lies on the y-axis" is false, since the center is (1,0) which lies on the x-axis.