Consider two points and in the coordinate plane. A point and a point in the same plane satisfy the following conditions:Let in triangle , and let the area of triangle be denoted by .For an angle satisfyingassume thatwhere and are relatively prime positive integers.(Also, point does not lie on segment , and point is distinct from point .)Find the value ofoindent
[THOUGHT] The position of C is (5 - 6cos(θ), 6sin(θ)). Parameter s = 4 / (1 + 5cos(θ)). The area f(θ) = 30 * sin(θ) * (1 - s) = 30 * sin(θ) * (5cos(θ) - 3) / (1 + 5*cos(θ)). To verify f'(β) where cos(β) = 4/5, sin(β) = 3/5, use sympy to compute the derivative and evaluate. [/THOUGHT] [PYTHON] import sympy as sp
theta = sp.symbols('theta') c = sp.cos(theta) s = sp.sin(theta) f = 30 * s * (5c - 3) / (1 + 5c)
df = sp.diff(f, theta) beta_val = df.subs({c: 4/5, s: 3/5}) print(beta_val) print(sp.simplify(beta_val)) [/PYTHON]
[THOUGHT] The Python output -3.84 is -96/25, so p=96, q=25, p + q^2 = 96 + 625 = 721. [/THOUGHT] \boxed{721}
Explanation
The provided AI thought derived the function f(θ) = 30 * sin(θ) * (5cos(θ) - 3) / (1 + 5cos(θ)) based on the geometric setup: point C at (5 - 6cos(θ), 6sin(θ)), parameter s = 4 / (1 + 5*cos(θ)) for point P on AC, and area of triangle BCP as 30 * sin(θ) * (1 - s).
The derivative f'(θ) was computed using SymPy and evaluated at β where cos(β) = 4/5 and sin(β) = 3/5, yielding -3.84000000000000.
This numerical value equals -96/25 exactly, as 96 ÷ 25 = 3.84. Here, 96 and 25 are coprime positive integers, satisfying f'(β) = -p/q with p = 96 and q = 25.
Thus, p + q² = 96 + 25² = 96 + 625 = 721.