Question

determining whether a point lies on a circle. a circle centered at the origin contains the point (0, -9). does (8, √17) also lie on the circle? explain. no, the distance from the center to the point (8, √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, √17). yes, the distance from the origin to the point (8, √17) is 9 units. yes, the distance from the point (0, -9) to the point (8, √17) is 9 units.

Explanation:

Step1: Calculate the radius of the circle

The circle is centered at the origin $(0,0)$ and contains the point $(0, - 9)$. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, with $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(0,-9)$, 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 $d=\sqrt{(8 - 0)^2+(\sqrt{17}-0)^2}=\sqrt{64 + 17}=\sqrt{81}=9$.
Since the distance from the origin to the point $(8,\sqrt{17})$ is equal to the radius of the circle, the point lies on the circle.

Answer:

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