\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] To solve the problem, we start by determining the parametric equations for the positions of ( P_1 ) and ( P_2 ). Given the speed ratio ( |P_1'(t)| : |P_2'(t)| = 1 : 4 ), we find that ( \theta(t) = 2t ) for ( P_2 ). The distance squared between ( P_1 ) and ( P_2 ) is derived and simplified to a quadratic function in ( \cos t ). The maximum distance occurs when ( \cos t = -\frac{1}{8} ). Using this value, we compute the required trigonometric expression involving ( \sin t ), ( \cos t ), ( \sin \theta(t) ), and ( \cos \theta(t) ) at ( t = t_0 ). The expression simplifies to ( \frac{77}{64} ), leading to ( p + q = 141 ).
[PYTHON] import math from sympy import symbols, solve, cos, sin, simplify
Define symbols
t = symbols('t') u = cos(t)
Distance squared simplified to quadratic in u
D_squared = -8u**2 - 2u + 10
Find u that maximizes D_squared (vertex of parabola)
u_max = solve(D_squared.diff(u), u)[0] u_val = u_max.evalf() # Should be -1/8
Calculate sin(t) when cos(t) = -1/8
sin_t = sqrt(1 - u_val**2) sin_t = sin_t.evalf()
Calculate theta(t) = 2t, so cos(theta) and sin(theta)
cos_theta = 2u_val**2 - 1 sin_theta = 2sin_t*u_val
Compute the expression: sin(t)2sin(theta) - (1 + cos(t))2cos(theta)
expression = sin_t * 2 * sin_theta - (1 + u_val) * 2 * cos_theta expression_simplified = simplify(expression)
Since we know the exact value should be 77/64, verify with exact fractions
u_exact = -1/8 sin_t_exact = sqrt(1 - u_exact2) cos_theta_exact = 2*u_exact2 - 1 sin_theta_exact = 2sin_t_exactu_exact expression_exact = sin_t_exact * 2 * sin_theta_exact - (1 + u_exact) * 2 * cos_theta_exact numerator = expression_exact.as_numer_denom()[0] denominator = expression_exact.as_numer_denom()[1] p_plus_q = numerator + denominator
print(p_plus_q) [/PYTHON]
[THOUGHT] The error in the Python code was due to an incorrect derivative approach. Manually solving, the maximum distance occurs at ( \cos t = -\frac{1}{8} ). Calculating the expression with ( \sin t = \frac{3\sqrt{7}}{8} ), ( \cos \theta = -\frac{15}{16} ), and ( \sin \theta = -\frac{3\sqrt{7}}{32} ), the expression simplifies to ( \frac{147}{128} ). Thus, ( p + q = 147 + 128 = 275 ).
\boxed{275}