To determine whether a point lies on a given circle, we can follow these steps:
1. Determine the Radius of the Circle:
- Calculate the radius of the circle using the distance between the center of the circle (which is at the origin [tex]\( (0, 0) \)[/tex]) and the given point [tex]\( (0, -9) \)[/tex].
- Use the distance formula to find the radius:
[tex]\[
\text{Radius} = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}
\][/tex]
Plugging in the points [tex]\( (0, 0) \)[/tex] and [tex]\( (0, -9) \)[/tex]:
[tex]\[
\text{Radius} = \sqrt{(0 - 0)^2 + (-9 - 0)^2} = \sqrt{0 + 81} = \sqrt{81} = 9
\][/tex]
Therefore, the radius of the circle is 9 units.
2. Calculate the Distance from the Center to the Point [tex]\( (8, \sqrt{17}) \)[/tex]:
- Again, use the distance formula to calculate the distance from the origin [tex]\( (0, 0) \)[/tex] to the point [tex]\( (8, \sqrt{17}) \)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_0)^2 + (y_2 - y_0)^2}
\][/tex]
Plugging in the points [tex]\( (0, 0) \)[/tex] and [tex]\( (8, \sqrt{17}) \)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(8 - 0)^2 + (\sqrt{17} - 0)^2} = \sqrt{8^2 + (\sqrt{17})^2} = \sqrt{64 + 17} = \sqrt{81} = 9
\][/tex]
Therefore, the distance from the center to the point [tex]\( (8, \sqrt{17}) \)[/tex] is 9 units.
3. Compare the Distance to the Radius:
- Since the distance from the center to the point [tex]\( (8, \sqrt{17}) \)[/tex] is 9 units, which is equal to the radius of the circle, the point [tex]\( (8, \sqrt{17}) \)[/tex] lies on the circle.
Thus, the correct statement is:
- Yes, the distance from the origin to the point [tex]\( (8, \sqrt{17}) \)[/tex] is 9 units.
