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
The area function is given by (f(\theta) = 30 \sin \theta \cdot \frac{5 \cos \theta - 3}{5 \cos \theta + 1}). The derivative (f'(\theta)) is computed, and evaluated at (\theta = \beta) where (\cos \beta = \frac{4}{5}) and (\sin \beta = \frac{3}{5}).
After differentiation and substitution, (f'(\beta) = -\frac{96}{25}). Thus, (-\frac{p}{q} = -\frac{96}{25}), so (\frac{p}{q} = \frac{96}{25}). The integers 96 and 25 are relatively prime, so (p = 96) and (q = 25).
Then, (p + q^2 = 96 + 25^2 = 96 + 625 = 721).
\boxed{721}
[THOUGHT] We are given cosβ = 4/5, so sinβ = 3/5. We need to compute f'(β) where f(θ) = 30 sinθ (5 cosθ - 3) / (5 cosθ + 1). We will use sympy to compute the derivative and evaluate at β by substituting the values of cosβ and sinβ. [/THOUGHT] [PYTHON] from sympy import symbols, sin, cos, diff theta = symbols('theta') f = 30 * sin(theta) * (5cos(theta) - 3) / (5cos(theta) + 1) f_prime = diff(f, theta) f_prime_val = f_prime.subs({cos(theta): 4/5, sin(theta): 3/5}) print(f_prime_val) [/PYTHON]
\boxed{721}