To determine which point lies on a circle with a radius of 5 units and center at [tex]\( P(6,1) \)[/tex], we need to calculate the distance of each given point from the center. If the distance from a point to the center is exactly equal to the radius, then the point lies on the circle.
The formula for the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
We are given the center [tex]\( P(6,1) \)[/tex] and the radius [tex]\( r = 5 \)[/tex]. We'll calculate the distance for each point from the center to see if it equals the radius.
1. Point [tex]\( Q(1,11) \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(6 - 1)^2 + (1 - 11)^2} \][/tex]
[tex]\[ = \sqrt{5^2 + (-10)^2} \][/tex]
[tex]\[ = \sqrt{25 + 100} \][/tex]
[tex]\[ = \sqrt{125} \][/tex]
[tex]\[ = 11.180 \][/tex]
2. Point [tex]\( R(2,4) \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(6 - 2)^2 + (1 - 4)^2} \][/tex]
[tex]\[ = \sqrt{4^2 + (-3)^2} \][/tex]
[tex]\[ = \sqrt{16 + 9} \][/tex]
[tex]\[ = \sqrt{25} \][/tex]
[tex]\[ = 5.0 \][/tex]
3. Point [tex]\( S(4,-4) \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(6 - 4)^2 + (1 - (-4))^2} \][/tex]
[tex]\[ = \sqrt{2^2 + 5^2} \][/tex]
[tex]\[ = \sqrt{4 + 25} \][/tex]
[tex]\[ = \sqrt{29} \][/tex]
[tex]\[ = 5.385 \][/tex]
4. Point [tex]\( T(9,-2) \)[/tex]:
[tex]\[ \text{Distance} = \sqrt{(6 - 9)^2 + (1 - (-2))^2} \][/tex]
[tex]\[ = \sqrt{(-3)^2 + 3^2} \][/tex]
[tex]\[ = \sqrt{9 + 9} \][/tex]
[tex]\[ = \sqrt{18} \][/tex]
[tex]\[ = 4.243 \][/tex]
From the calculations:
- The distance for [tex]\( Q(1,11) \)[/tex] is [tex]\( 11.180 \)[/tex], which is not equal to the radius.
- The distance for [tex]\( R(2,4) \)[/tex] is exactly [tex]\( 5.0 \)[/tex], which equals the radius.
- The distance for [tex]\( S(4,-4) \)[/tex] is [tex]\( 5.385 \)[/tex], which is not equal to the radius.
- The distance for [tex]\( T(9,-2) \)[/tex] is [tex]\( 4.243 \)[/tex], which is not equal to the radius.
Therefore, the correct answer is (B) [tex]\( R(2,4) \)[/tex].
