To determine if a given point lies on the circle centered at the origin with point [tex]\( Q(10,0) \)[/tex] on it, we first identify the radius:
Since [tex]\( Q(10, 0) \)[/tex] lies on the circle, the radius [tex]\( r \)[/tex] of the circle starting from the origin [tex]\((0, 0)\)[/tex] is 10 units, because:
[tex]\[ r = \sqrt{(10 - 0)^2 + (0 - 0)^2} = \sqrt{10^2} = 10 \][/tex]
The equation of the circle centered at the origin with radius 10 is:
[tex]\[ x^2 + y^2 = 10^2 = 100 \][/tex]
Now, we need to check if each of the given points lies on the circle by confirming if they satisfy the equation.
Option A: [tex]\((3, \sqrt{34})\)[/tex]
[tex]\[ x = 3, \, y = \sqrt{34} \][/tex]
Substitute these values into the circle equation:
[tex]\[ 3^2 + (\sqrt{34})^2 = 9 + 34 = 43 \][/tex]
Since 43 is not equal to 100, point A does not lie on the circle.
Option B: [tex]\((4.5, \sqrt{3})\)[/tex]
[tex]\[ x = 4.5, \, y = \sqrt{3} \][/tex]
Substitute these values into the circle equation:
[tex]\[ 4.5^2 + (\sqrt{3})^2 = 20.25 + 3 = 23.25 \][/tex]
Since 23.25 is not equal to 100, point B does not lie on the circle.
Option C: [tex]\((6, 4)\)[/tex]
[tex]\[ x = 6, \, y = 4 \][/tex]
Substitute these values into the circle equation:
[tex]\[ 6^2 + 4^2 = 36 + 16 = 52 \][/tex]
Since 52 is not equal to 100, point C does not lie on the circle.
Option D: [tex]\((5, 5\sqrt{3})\)[/tex]
[tex]\[ x = 5, \, y = 5\sqrt{3} \][/tex]
Substitute these values into the circle equation:
[tex]\[ 5^2 + (5\sqrt{3})^2 = 25 + 75 = 100 \][/tex]
Since 100 is equal to 100, point D lies on the circle.
Therefore, the correct answer is:
D. [tex]\((5, 5\sqrt{3})\)[/tex]
