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

Explanation:

Standard form for a circle is ( x -h)^2 + ( y-k)^2 = r^2

h, k is the center r is the radius

x^2 -2x + y^2 = 8 arrange into standard form ( complete the square for 'x')

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

(x-1)^2 + y^2 = 9 shows center is at 1, 0

radius = 3

So

The radius of the circle is 3 units.

The center of the circle lies on the x-axis (1,0 is on the x axis)

The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

Answer 2

The three true statements are

• A) The radius of the circle is 3 units
• B) The center of the circle lies on the x-axis
• E) The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

EXPLANATION :

1. To see why A) is true, we can rearrange the equation to get it in the form (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. Completing the square, we have x² - 2x + y² - 8 = 0, which can be rewritten as (x - 1)² - 1 + y² - 8 = 0. Rearranging again, we get (x - 1)² + y² = 9, which is the equation of a circle with center (1, 0) and radius 3.
2. To see why B) is true, note that the x-coordinate of the center of the circle is given by x = 1, so it lies on the x-axis.
3. To see why E) is true, note that the equation x² + y² = 9 is the equation of a circle with center (0, 0) and radius 3. We can rewrite the equation of the given circle as (x - 1)² + y² = 9, which is the equation of a circle with the same radius and a center shifted to (1, 0).