Answer :
To determine which of the given points lies on the circle centered at the origin with a radius of 5 units, we need to calculate the distance of each point from the center and see if it is equal to the radius. The distance formula is given by:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Since the circle is centered at the origin (0, 0), the formula simplifies to:
[tex]\[ \text{Distance} = \sqrt{x^2 + y^2} \][/tex]
Let's apply this formula to each of the points provided:
### 1. Point (2, \(\sqrt{21}\))
[tex]\[ \text{Distance} = \sqrt{(2 - 0)^2 + (\sqrt{21} - 0)^2} = \sqrt{2^2 + (\sqrt{21})^2} = \sqrt{4 + 21} = \sqrt{25} = 5 \][/tex]
The distance is exactly 5 units, so this point is on the circle.
### 2. Point (2, \(\sqrt{23}\))
[tex]\[ \text{Distance} = \sqrt{(2 - 0)^2 + (\sqrt{23} - 0)^2} = \sqrt{2^2 + (\sqrt{23})^2} = \sqrt{4 + 23} = \sqrt{27} \approx 5.196 \][/tex]
The distance is approximately 5.196 units, which is not equal to the radius.
### 3. Point (2, 1)
[tex]\[ \text{Distance} = \sqrt{(2 - 0)^2 + (1 - 0)^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \][/tex]
The distance is approximately 2.236 units, which is not equal to the radius.
### 4. Point (2, 3)
[tex]\[ \text{Distance} = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.606 \][/tex]
The distance is approximately 3.606 units, which is not equal to the radius.
### Conclusion
Among the given points, only the point (2, [tex]\(\sqrt{21}\)[/tex]) has a distance exactly equal to the radius of 5 units. Therefore, the point [tex]\((2, \sqrt{21})\)[/tex] is on the circle centered at the origin with a radius of 5 units.
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Since the circle is centered at the origin (0, 0), the formula simplifies to:
[tex]\[ \text{Distance} = \sqrt{x^2 + y^2} \][/tex]
Let's apply this formula to each of the points provided:
### 1. Point (2, \(\sqrt{21}\))
[tex]\[ \text{Distance} = \sqrt{(2 - 0)^2 + (\sqrt{21} - 0)^2} = \sqrt{2^2 + (\sqrt{21})^2} = \sqrt{4 + 21} = \sqrt{25} = 5 \][/tex]
The distance is exactly 5 units, so this point is on the circle.
### 2. Point (2, \(\sqrt{23}\))
[tex]\[ \text{Distance} = \sqrt{(2 - 0)^2 + (\sqrt{23} - 0)^2} = \sqrt{2^2 + (\sqrt{23})^2} = \sqrt{4 + 23} = \sqrt{27} \approx 5.196 \][/tex]
The distance is approximately 5.196 units, which is not equal to the radius.
### 3. Point (2, 1)
[tex]\[ \text{Distance} = \sqrt{(2 - 0)^2 + (1 - 0)^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \][/tex]
The distance is approximately 2.236 units, which is not equal to the radius.
### 4. Point (2, 3)
[tex]\[ \text{Distance} = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.606 \][/tex]
The distance is approximately 3.606 units, which is not equal to the radius.
### Conclusion
Among the given points, only the point (2, [tex]\(\sqrt{21}\)[/tex]) has a distance exactly equal to the radius of 5 units. Therefore, the point [tex]\((2, \sqrt{21})\)[/tex] is on the circle centered at the origin with a radius of 5 units.
