Question

a. definition of a circle
b. distance formula
c. definition of radius
d. definition of diameter

Explanation:

To determine if the point \((1,\sqrt{3})\) lies on the circle centered at the origin \((0,0)\) and passing through \((0,2)\), we use the distance formula. The radius of the circle is the distance from the origin \((0,0)\) to \((0,2)\), which is \(r = \sqrt{(0 - 0)^2+(2 - 0)^2}=2\). Then we check the distance from the origin \((0,0)\) to \((1,\sqrt{3})\) using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Substituting \(x_1 = 0,y_1 = 0,x_2 = 1,y_2=\sqrt{3}\), we get \(d=\sqrt{(1 - 0)^2+(\sqrt{3}-0)^2}=\sqrt{1 + 3}=\sqrt{4}=2\). Since the distance from the origin to \((1,\sqrt{3})\) is equal to the radius (found using the distance formula for the point \((0,2)\)), the distance formula is used to verify the point lies on the circle.

Answer:

B. Distance formula