[THOUGHT] We need to parse the problem carefully.

We have points A(5,0) and B(-5,0). C is a point such that AC = 6. P is a point on segment AC (so P lies between A and C). Condition (iii): BP + CP = 8.

Let angle A in triangle ABC be θ (i.e., angle at vertex A between AB and AC). So θ = ∠CAB.

We need to define f(θ) = area of triangle BCP. Then we are told that for β such that cos β = 4/5, 0 < β < π/2, we have f'(β) = -p/q with p,q coprime positive integers. Find p + q^2.

We must assume point C does not lie on segment AB (so C is not collinear with A and B, and not between them), and P distinct from C.

We need to derive f(θ) as a function of θ, differentiate, evaluate at β where cos β = 4/5, find derivative value, then compute p+q^2.

Let's set up coordinates.

Let A = (5,0), B = (-5,0). Let C be such that AC = 6. Let’s parameterize C using angle at A: θ = ∠CAB. That is angle between AB and AC. AB vector from A to B: B - A = (-10, 0). So AB is along negative x-axis from A. Alternatively, angle A is measured from AB to AC. Usually angle at vertex A is between AB and AC. Since AB is from A to B, direction leftwards. So we can define AC direction: from A, length 6, making angle θ with AB. But careful: angle A is interior angle of triangle ABC. AB is one side, AC is another. So vector AB = B - A = (-10,0). Vector AC = C - A. The angle between AB and AC is θ.

Thus we can write C = A + 6*(cos(π - θ), sin(π - θ))? Wait: AB direction is from A to B: direction angle π (180°) from positive x-axis. AC direction from A is some angle φ such that angle between AB and AC is θ. So φ = π - θ (if θ measured from AB to AC in counterclockwise direction?). Actually, we need to decide orientation. Since C is above or below the x-axis? The problem says C does not lie on segment AB, but could be above or below. Without loss, we can assume C is above x-axis (positive y). Then AB direction is leftwards (π). AC direction is at angle π - θ (since angle between leftwards direction and AC direction is θ). So AC vector = 6*(cos(π - θ), sin(π - θ)) = 6*(-cos θ, sin θ). That gives C = A + AC = (5 - 6 cos θ, 6 sin θ). Check: when θ=0, C = (5 - 6, 0) = (-1,0) which is on x-axis left of A but not between A and B? Actually (-1,0) is between A(5,0) and B(-5,0)? No, because B is at -5, so -1 is between -5 and 5, so C would be on segment AB. But condition says C does not lie on segment AB. So θ cannot be 0. For θ=π/2, C = (5,6). That's fine.

Thus C = (5 - 6 cos θ, 6 sin θ). Let's verify distance AC = 6: (Δx)^2 + (Δy)^2 = (-6 cos θ)^2 + (6 sin θ)^2 = 36(cos^2 θ + sin^2 θ) = 36, so length = 6. Good.

Now triangle ABC: vertices A(5,0), B(-5,0), C(5-6 cos θ, 6 sin θ). Angle at A is θ by construction.

Now P lies on segment AC. Let P = A + t*(C - A) for some t in (0,1) (since P distinct from A and C, t in (0,1)). So P = (5 - 6t cos θ, 6t sin θ). Because C - A = (-6 cos θ, 6 sin θ).

Condition (iii): BP + CP = 8.

We need to compute distances BP and CP.

Let’s compute coordinates:

B = (-5,0) P = (5 - 6t cos θ, 6t sin θ) C = (5 - 6 cos θ, 6 sin θ)

Compute BP = distance between B and P: Δx = (5 - 6t cos θ) - (-5) = 10 - 6t cos θ Δy = 6t sin θ - 0 = 6t sin θ So BP = sqrt((10 - 6t cos θ)^2 + (6t sin θ)^2).

Compute CP = distance between C and P: Δx = (5 - 6 cos θ) - (5 - 6t cos θ) = -6 cos θ + 6t cos θ = -6 cos θ (1 - t) = -6 cos θ (1 - t) Δy = 6 sin θ - 6t sin θ = 6 sin θ (1 - t) So CP = sqrt( (6 cos θ (1 - t))^2 + (6 sin θ (1 - t))^2 ) = 6(1 - t) sqrt(cos^2 θ + sin^2 θ) = 6(1 - t).

Thus CP = 6(1 - t). Interesting.

Now BP = sqrt( (10 - 6t cos θ)^2 + (36 t^2 sin^2 θ) ).

Simplify BP^2: (10 - 6t cos θ)^2 + 36 t^2 sin^2 θ = 100 - 120 t cos θ + 36 t^2 cos^2 θ + 36 t^2 sin^2 θ = 100 - 120 t cos θ + 36 t^2 (cos^2 θ + sin^2 θ) = 100 - 120 t cos θ + 36 t^2.

Thus BP = sqrt(100 - 120 t cos θ + 36 t^2).

Now condition (iii): BP + CP = 8 => sqrt(100 - 120 t cos θ + 36 t^2) + 6(1 - t) = 8.

Let’s denote u = t. Then equation:

sqrt(100 - 120 u cos θ + 36 u^2) = 8 - 6(1 - u) = 8 - 6 + 6u = 2 + 6u.

Thus sqrt(100 - 120 u cos θ + 36 u^2) = 2 + 6u.

Square both sides:

100 - 120 u cos θ + 36 u^2 = (2 + 6u)^2 = 4 + 24u + 36 u^2.

Cancel 36 u^2 from both sides:

100 - 120 u cos θ = 4 + 24u.

Thus 100 - 120 u cos θ - 4 - 24u = 0 => 96 - 120 u cos θ - 24u = 0 => divide 24: 4 - 5 u cos θ - u = 0 => 4 - u(5 cos θ + 1) = 0.

Thus u = t = 4 / (5 cos θ + 1).

Check: u must be between 0 and 1. Since cos θ ∈ [-1,1], denominator ∈ [ -5+1, 5+1] = [-4,6]. But u positive, so denominator must be positive. Also u < 1. We'll assume θ in (0,π) such that denominator > 0. For θ in (0,π/2), cos θ > 0, denominator > 1, so u < 4/(1) = 4? Wait: denominator min positive? Actually if cos θ > 0, denominator > 1, so u < 4. But u must be <1. So we need 4/(5 cos θ + 1) < 1 => 4 < 5 cos θ + 1 => 5 cos θ > 3 => cos θ > 3/5. So θ < arccos(3/5). That's consistent with later β where cos β = 4/5 > 3/5. So indeed θ is less than arccos(3/5). Good.

Now we have t expressed in terms of θ.

Now we need area of triangle BCP, f(θ). Points B(-5,0), C(5 - 6 cos θ, 6 sin θ), P(5 - 6t cos θ, 6t sin θ). Area can be computed using shoelace formula.

Let’s compute area = 1/2 | x_B(y_C - y_P) + x_C(y_P - y_B) + x_P(y_B - y_C) |.

Since points are not collinear (presumably), area positive.

Compute:

x_B = -5, y_B = 0. x_C = 5 - 6 cos θ, y_C = 6 sin θ. x_P = 5 - 6t cos θ, y_P = 6t sin θ.

Compute differences:

y_C - y_P = 6 sin θ - 6t sin θ = 6 sin θ (1 - t). y_P - y_B = 6t sin θ - 0 = 6t sin θ. y_B - y_C = 0 - 6 sin θ = -6 sin θ.

Now compute sum:

Term1 = x_B * (y_C - y_P) = (-5) * 6 sin θ (1 - t) = -30 sin θ (1 - t). Term2 = x_C * (y_P - y_B) = (5 - 6 cos θ) * 6t sin θ = 6t sin θ (5 - 6 cos θ). Term3 = x_P * (y_B - y_C) = (5 - 6t cos θ) * (-6 sin θ) = -6 sin θ (5 - 6t cos θ).

Sum = -30 sin θ (1 - t) + 6t sin θ (5 - 6 cos θ) - 6 sin θ (5 - 6t cos θ).

Factor sin θ (assuming sin θ > 0, which is fine for θ in (0,π) except 0,π; we can take absolute value later).

Sum = sin θ [ -30(1 - t) + 6t(5 - 6 cos θ) - 6(5 - 6t cos θ) ].

Simplify inside:

First expand: -30 + 30t + 6t*5 - 36t cos θ - 30 + 36t cos θ.

Wait carefully:

-30(1 - t) = -30 + 30t. 6t(5 - 6 cos θ) = 30t - 36t cos θ. -6(5 - 6t cos θ) = -30 + 36t cos θ.

Now sum: (-30 + 30t) + (30t - 36t cos θ) + (-30 + 36t cos θ) = combine constants: -30 -30 = -60. Combine t terms: 30t + 30t = 60t. Combine t cos θ terms: -36t cos θ + 36t cos θ = 0.

Thus sum = sin θ (60t - 60) = 60 sin θ (t - 1).

So area = 1/2 * |60 sin θ (t - 1)| = 30 |sin θ (t - 1)|.

Since t < 1 (P between A and C, not at C), t - 1 is negative. So (t - 1) = -(1 - t). Thus |sin θ (t - 1)| = sin θ (1 - t) (since sin θ > 0 for θ in (0,π)). So area f(θ) = 30 sin θ (1 - t).

Thus f(θ) = 30 sin θ (1 - t).

Now substitute t = 4/(5 cos θ + 1). Then 1 - t = 1 - 4/(5 cos θ + 1) = (5 cos θ + 1 - 4)/(5 cos θ + 1) = (5 cos θ - 3)/(5 cos θ + 1).

Thus f(θ) = 30 sin θ * (5 cos θ - 3)/(5 cos θ + 1).

So f(θ) = 30 sin θ (5 cos θ - 3) / (5 cos θ + 1).

Now we need derivative f'(θ). Compute derivative with respect to θ.

Let’s denote numerator N(θ) = 30 sin θ (5 cos θ - 3). Denominator D(θ) = (5 cos θ + 1). Then f(θ) = N/D.

We can compute derivative using quotient rule.

But first, we need to evaluate f'(β) where cos β = 4/5, 0<β<π/2.

We can compute derivative analytically or numerically using Python to verify.

Let's compute f'(θ) = (N' D - N D') / D^2.

Compute N = 30 sin θ (5 cos θ - 3) = 30 [5 sin θ cos θ - 3 sin θ] = 30 [ (5/2) sin 2θ - 3 sin θ ]? Might be easier to differentiate directly.

Let’s compute derivative step by step with Python.

We'll compute f(θ) as defined, differentiate symbolically or numerically.

We'll use sympy to differentiate.

But careful: f(θ) expression derived might have domain restrictions: denominator 5 cos θ + 1 > 0 for t positive? Actually t = 4/(5 cos θ + 1) > 0 requires denominator positive. Also t < 1 gave cos θ > 3/5. So domain for θ is such that cos θ > 3/5, i.e., θ ∈ (0, arccos(3/5)). At β, cos β = 4/5 > 3/5, so β is in that interval. So fine.

Now compute f'(β) and then find p,q such that f'(β) = -p/q with p,q coprime positive integers. Then compute p + q^2.

Let's implement.

We'll compute derivative symbolically and evaluate at cos β = 4/5, sin β = sqrt(1 - (4/5)^2) = sqrt(1 - 16/25) = sqrt(9/25) = 3/5 (positive). So sin β = 3/5.

We'll compute f'(β) exactly as rational expression.

Let's do Python. </think>

[PYTHON] import sympy as sp

Define variable theta

theta = sp.symbols('theta')

Define f(theta) = 30 * sin(theta) * (5cos(theta) - 3) / (5cos(theta) + 1)

f = 30 * sp.sin(theta) * (5sp.cos(theta) - 3) / (5sp.cos(theta) + 1)

Compute derivative

f_prime = sp.diff(f, theta) sp.simplify(f_prime) [/PYTHON]

[THOUGHT] We need to compute ( f'(\beta) ) where (\cos \beta = \frac{4}{5}), (0 < \beta < \frac{\pi}{2}), and (f'(\beta) = -\frac{p}{q}) with (p,q) coprime positive integers. Then compute (p + q^2).

From the Python output, (f'(\theta) = \frac{30\left( \frac{63}{4} \cos\theta + 5\cos 2\theta + \frac{25}{4} \cos 3\theta - 15 \right)}{(5\cos\theta + 1)^2}).

Given (\cos\beta = \frac{4}{5}), we compute (\sin\beta = \sqrt{1 - (16/25)} = \sqrt{9/25} = \frac{3}{5}).

Now compute (\cos 2\beta = 2\cos^2\beta - 1 = 2\cdot \frac{16}{25} - 1 = \frac{32}{25} - 1 = \frac{7}{25}).

(\cos 3\beta = 4\cos^3\beta - 3\cos\beta = 4\cdot \frac{64}{125} - 3\cdot \frac{4}{5} = \frac{256}{125} - \frac{12}{5} = \frac{256}{125} - \frac{300}{125} = -\frac{44}{125}).

Plug into numerator: (\frac{63}{4} \cdot \frac{4}{5} + 5 \cdot \frac{7}{25} + \frac{25}{4} \cdot \left(-\frac{44}{125}\right) - 15).

Compute each term:

1. (\frac{63}{4} \cdot \frac{4}{5} = \frac{63}{5} = 12.6) (keep as fraction: (63/5)).

2. (5 \cdot \frac{7}{25} = \frac{35}{25} = \frac{7}{5}).

3. (\frac{25}{4} \cdot \left(-\frac{44}{125}\right) = -\frac{25\cdot 44}{4\cdot 125} = -\frac{44}{4\cdot 5} = -\frac{44}{20} = -\frac{11}{5}).

4. (-15 = -\frac{75}{5}).

Sum: (\frac{63}{5} + \frac{7}{5} - \frac{11}{5} - \frac{75}{5} = \frac{63+7-11-75}{5} = \frac{-16}{5}).

So numerator = (-16/5).

Denominator: ((5\cos\beta + 1)^2 = \left(5\cdot \frac{4}{5} + 1\right)^2 = (4 + 1)^2 = 25).

Thus (f'(\beta) = \frac{30 \cdot (-16/5)}{25} = \frac{30 \cdot (-16)}{5 \cdot 25} = \frac{480}{125} \cdot (-1)??) Wait carefully: (30 \cdot (-16/5) = -480/5 = -96). Then (-96 / 25 = -\frac{96}{25}).

So (f'(\beta) = -\frac{96}{25}). Thus (p = 96), (q = 25), (p+q^2 = 96 + 625 = 721).

Check: (p,q) coprime? gcd(96,25)=1 yes.

Final answer: 721. </think>

\boxed{721}