- Calculate the squared radius of the circle using the point \((0, -9)\), which gives \(r^2 = 0^2 + (-9)^2 = 81\).
- Calculate the squared distance of the point \((8, \sqrt{17})\) from the origin, which gives \(8^2 + (\sqrt{17})^2 = 64 + 17 = 81\).
- Compare the distance of the point to the origin with the radius: since \(d^2 = r^2 = 81\), the point lies on the circle.
- The correct choice is: \(\boxed{\text{Yes, the distance from the origin to the point (8, \sqrt{17}) is 9 units.}}\)
Explanation
- Determine the circle's equation.
The problem asks to determine if the point \((8, \sqrt{17})\) lies on a circle centered at the origin \((0, 0)\) that passes through the point \((0, -9)\).
A circle centered at the origin with radius \(r\) consists of all points \((x, y)\) such that \(x^2 + y^2 = r^2\).
First, we find the radius squared (\(r^2\)) using the point \((0, -9)\):
\[r^2 = 0^2 + (-9)^2 = 81\]
Thus, the radius is \(r = \sqrt{81} = 9\). The equation of our circle is \(x^2 + y^2 = 81\).
- Check if the point lies on the circle.
Now, we check if the point \((8, \sqrt{17})\) satisfies the circle's equation \(x^2 + y^2 = 81\).
We calculate the squared distance of the point \((8, \sqrt{17})\) from the origin:
\[d^2 = x^2 + y^2 = 8^2 + (\sqrt{17})^2\]
\[d^2 = 64 + 17\]
\[d^2 = 81\]
Since \(d^2 = 81\), the distance \(d = \sqrt{81} = 9\), which is equal to the radius of the circle. Thus, the point \((8, \sqrt{17})\) lies on the circle.
- Conclusion.
Comparing our result with the given choices:
- "No, the distance from the center to the point \((8, \sqrt{17})\) is not the same as the radius." (False)
- "No, the radius of 10 units is different from the distance from the center to the point \((8, \sqrt{17})\)." (False)
- "Yes, the distance from the origin to the point \((8, \sqrt{17})\) is 9 units." (True)
- "Yes, the distance from the point \((0, -9)\) to the point \((8, \sqrt{17})\) is 9 units." (False)
The correct option is: Yes, the distance from the origin to the point \((8, \sqrt{17})\) is 9 units.
Examples
Think of a circular radar screen centered at your location (the origin). If a target at coordinates \((0, -9)\) is detected on the edge of your radar, the radius of detection is 9 units. If you then detect another target at \((8, \sqrt{17})\), you can use the same distance formula to see if it is at the same distance, 9 units, confirming it is also on the edge of your radar range. This helps in tracking objects at a constant distance from a fixed point.