[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, $p + q = 147 + 128 = 275$.

$\boxed{275}$