\text{Let } P_1(1+\cos t,; \sin t)\text{ be a point moving on the circle with center } (1,0)\text{ and radius } 1,]Both points start at the intersection point of the two circlesand move counterclockwise along their respective circles.Assume that their speeds satisfyLet be the time when the distance between and is maximized.At this moment, the following expression:where and are relatively prime positive integers.Find the value of .oindent
[THOUGHT] The critical points occur where cos(t) = -1/8. At these points, the expression simplifies to 77/64 independently of the sign of sin(t). To verify, use SymPy to compute the expression exactly.
Define cos_t = -1/8, sin2_t = 2 * sin_t * cos_t but use identities: cos(2t) = 2*cos_t2 - 1 = -31/32, sin^2(t) = 1 - cos_t2 = 63/64. Then compute the expression. [/THOUGHT] [PYTHON] from sympy import *
cos_t = Rational(-1,8) sin2_t_over_sin_t = 2 * cos_t # sin(2t)/sin(t) = 2 cos t cos_2t = 2*cos_t2 - 1 sin_t_sq = 1 - cos_t2 one_plus_cos_t = 1 + cos_t
inside = sin_t_sq * (sin2_t_over_sin_t)/2 * (-1) - one_plus_cos_t * cos_2t # wait, better direct
Actually: sin_t * sin_2t = sin_t * 2 sin_t cos_t = 2 cos_t sin_t^2
No: expression inside /1: sin t sin 2t - (1+cos t) cos 2t = sin t (2 sin t cos t) - (1+c) cos2t = 2 cos t sin^2 t - (1+c) cos2t
expr_inside = 2 * cos_t * sin_t_sq - one_plus_cos_t * cos_2t
g = 2 * expr_inside
print(g) print(N(g)) # numerical check [/PYTHON]
[THOUGHT] The Python output confirms the expression equals 77/64, where p=77 and q=64 are coprime. Thus, p+q=141. [/THOUGHT] \boxed{141}
Explanation
The reasoning process begins by recognizing that the maximum distance between points P1 and P2 occurs at critical points where the derivative condition leads to cos(t) = -1/8. At this value, the expression sin(t) · 2 sin(θ(t)) - (1 + cos(t)) · 2 cos(θ(t)) simplifies to a constant value.
To compute this, substitute cos(t) = -1/8 into relevant trigonometric identities:
- sin²(t) = 1 - cos²(t) = 1 - (-1/8)² = 1 - 1/64 = 63/64.
- cos(2t) = 2 cos²(t) - 1 = 2(-1/8)² - 1 = 2(1/64) - 1 = 2/64 - 1 = 1/32 - 1 = -31/32.
- 1 + cos(t) = 1 + (-1/8) = 7/8.
The expression expands as follows, using the relation derived from the speed ratio and distance maximization:
- sin(t) · 2 sin(θ(t)) - (1 + cos(t)) · 2 cos(θ(t)) corresponds to 2 [cos(t) sin²(t) - (1 + cos(t)) cos(2t)], based on the geometric and kinematic constraints.
Substitute the values:
- First term: 2 cos(t) sin²(t) = 2 (-1/8) (63/64) = (-1/4) (63/64) = -63/256.
- Second term: (1 + cos(t)) cos(2t) = (7/8) (-31/32) = -217/256.
- Overall: 2 [ -63/256 - (-217/256) ] = 2 [ -63/256 + 217/256 ] = 2 (154/256) = 2 (77/128) = 77/64.
The Python code verifies this exact computation, outputting 77/64. Thus, p = 77 and q = 64 are coprime positive integers, yielding p + q = 141.