Let's solve the problem step-by-step:
Given:
1. The center of the circle is at the origin [tex]\((0, 0)\)[/tex].
2. A point on the circle is [tex]\((0, -9)\)[/tex].
3. We need to determine if the point [tex]\((8, \sqrt{17})\)[/tex] lies on the circle.
### Step 1: Calculate the radius of the circle
The radius of the circle is the distance from the origin to the point [tex]\((0, -9)\)[/tex]. Using the distance formula [tex]\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)[/tex]:
[tex]\[ \text{Radius} = \sqrt{(0 - 0)^2 + (-9 - 0)^2} = \sqrt{0 + 81} = \sqrt{81} = 9 \][/tex]
### Step 2: Calculate the distance from the origin to the point [tex]\((8, \sqrt{17})\)[/tex]
Using the distance formula again:
[tex]\[ d = \sqrt{(8 - 0)^2 + (\sqrt{17} - 0)^2} = \sqrt{8^2 + (\sqrt{17})^2} = \sqrt{64 + 17} = \sqrt{81} = 9 \][/tex]
### Step 3: Compare the distances
- The radius of the circle is 9.
- The distance from the origin to the point [tex]\((8, \sqrt{17})\)[/tex] is also 9.
Since these distances are equal, the point [tex]\((8, \sqrt{17})\)[/tex] lies on the circle.
### Conclusion
The correct statement is:
- Yes, the distance from the origin to the point [tex]\((8, \sqrt{17})\)[/tex] is 9 units.
So, the third option is correct:
- Yes, the distance from the origin to the point (8,√17) is 9 units.
