To determine which of the given points lies on the circle centered at the origin (0, 0) with a radius of 5 units, we need to calculate the distances of the points from the origin and see if any point's distance is exactly 5.
The distance formula used is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, the center of the circle (x_1, y_1) is (0, 0), so the distance formula simplifies to:
[tex]\[ d = \sqrt{x_2^2 + y_2^2} \][/tex]
We will calculate the distance for each of the given points:
1. For the point [tex]\((2, \sqrt{21})\)[/tex]:
[tex]\[ d = \sqrt{2^2 + (\sqrt{21})^2} = \sqrt{4 + 21} = \sqrt{25} = 5 \][/tex]
This point has a distance of 5 units from the origin.
2. For the point [tex]\((2, \sqrt{23})\)[/tex]:
[tex]\[ d = \sqrt{2^2 + (\sqrt{23})^2} = \sqrt{4 + 23} = \sqrt{27} \approx 5.196 \][/tex]
This point does not lie 5 units from the origin.
3. For the point [tex]\((2, 1)\)[/tex]:
[tex]\[ d = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.24 \][/tex]
This point does not lie 5 units from the origin.
4. For the point [tex]\((2, 3)\)[/tex]:
[tex]\[ d = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61 \][/tex]
This point does not lie 5 units from the origin.
Based on the calculations, the only point that is exactly 5 units away from the origin is [tex]\((2, \sqrt{21})\)[/tex].
Therefore, the point [tex]\((2, \sqrt{21})\)[/tex] is on the circle centered at the origin with a radius of 5 units.
