Question

a circle centered at the origin contains the point $(0,-9)$. does $(8,\sqrt{17})$ also lie on the circle? explain.
no, the distance from the center to the point $(8,\sqrt{17})$ is not the same as the radius.
no, the radius of 10 units is different from the distance from the center to the point $(8,\sqrt{17})$.
yes, the distance from the origin to the point $(8,\sqrt{17})$ is 9 units.
yes, the distance from the point $(0,-9)$ to the point $(8,\sqrt{17})$ is 9 units.

Explanation:

Step1: Calculate the radius

The radius \( r \) of the circle is the distance from the origin \((0,0)\) to the point \((0, - 9)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(r=\sqrt{(0 - 0)^2+(-9 - 0)^2}=\sqrt{0 + 81}=9\)

Step2: Calculate the distance from the origin to the point \((8,\sqrt{17})\)

Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), with \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(8,\sqrt{17})\), we get \(d=\sqrt{(8 - 0)^2+(\sqrt{17}-0)^2}=\sqrt{64 + 17}=\sqrt{81}=9\)

Answer:

Yes, the distance from the origin to the point \((8,\sqrt{17})\) is 9 units.