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: Find circle radius

The radius $r$ is the distance from origin $(0,0)$ to $(0,-9)$. Using distance formula $r=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$:
$r=\sqrt{(0-0)^2+(-9-0)^2}=\sqrt{0+81}=9$

Step2: Calculate distance to $(8,\sqrt{17})$

Compute distance $d$ from $(0,0)$ to $(8,\sqrt{17})$:
$d=\sqrt{(8-0)^2+(\sqrt{17}-0)^2}=\sqrt{64+17}=\sqrt{81}=9$

Step3: Compare distance and radius

Since $d=r=9$, the point lies on the circle.

Answer:

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