To determine if the point [tex]\((8, \sqrt{17})\)[/tex] lies on the circle with a center at the origin and containing the point [tex]\((0, -9)\)[/tex], follow these steps:
1. Identify the Radius of the Circle:
- Given that the circle passes through the point [tex]\((0, -9)\)[/tex].
- The radius of the circle can be found by calculating the distance from the origin [tex]\((0, 0)\)[/tex] to the point [tex]\((0, -9)\)[/tex]:
[tex]\[
\text{Radius} = \sqrt{(0 - 0)^2 + (0 - (-9))^2} = \sqrt{0 + 81} = \sqrt{81} = 9
\][/tex]
2. Calculate the Distance from the Origin to the Point [tex]\((8, \sqrt{17})\)[/tex]:
- The distance from the center of the circle (the origin, [tex]\((0,0)\)[/tex]) to the point [tex]\((8, \(\sqrt{17})\)[/tex]) will tell us if the point lies on the circle.
[tex]\[
d = \sqrt{(8 - 0)^2 + (\sqrt{17} - 0)^2} = \sqrt{8^2 + (\sqrt{17})^2} = \sqrt{64 + 17} = \sqrt{81} = 9
\][/tex]
3. Compare the Distance to the Radius:
- The calculated distance from the origin to the point [tex]\((8, \sqrt{17})\)[/tex] is 9, which is equal to the radius of the circle.
Since the distance from the origin to the point [tex]\((8, \sqrt{17})\)[/tex] is equal to the radius of the circle (both are 9 units), [tex]\((8, \sqrt{17})\)[/tex] lies on the circle.
Therefore, the correct answer is:
- Yes, the distance from the origin to the point (8, [tex]\(\sqrt{17})\)[/tex] is 9 units.
