Answer:
the coordinates of point P are (15, 19 + 3 sqrt(3)).
Explanation:
Let's first find the center of the circle, which is (h, k). From the given information, we know that h = 12 and k = 16.
Next, we need to find the coordinates of the point P. We can do this using the formula for finding a point on a circle given its center and angle:
x = h + r cos(theta)
y = k + r sin(theta)
where r is the radius of the circle and theta is the angle between the horizontal axis and the line connecting the center of the circle to the point P.
Substituting the given values, we have:
x = 12 + 6 cos(pi/3)
y = 16 + 6 sin(pi/3)
Evaluating the trigonometric functions, we get:
x = 12 + 6 (1/2) = 15
y = 16 + 6 (sqrt(3)/2) = 19 + 3 sqrt(3)
Therefore, the coordinates of point P are (15, 19 + 3 sqrt(3)).