[THOUGHT] I need to find the point P where the parametric curve intersects the line y = 3x - 5e. First, I'll substitute the parametric equations into the line equation to solve for t. Then, I'll compute the derivatives dx/dt and dy/dt to find the slope dy/dx at that t. Finally, I'll express the slope in the form (aπ - b)/(π + c) and compute a + b + c.

[/THOUGHT] [PYTHON] import math from sympy import symbols, sin, cos, exp, pi, diff, simplify, Rational

Define symbol

t = symbols('t')

Parametric equations

x = exp(4t) * (1 + sin(pit)**2) y = exp(4t) * (1 - 3cos(pi*t)**2)

Derivatives

dx_dt = diff(x, t) dy_dt = diff(y, t)

Slope expression

slope_expr = simplify(dy_dt / dx_dt)

Substitute t = 1/4

t_val = Rational(1, 4) slope_at_1_4 = simplify(slope_expr.subs(t, t_val))

Compute numerical value for verification

slope_num = float(slope_at_1_4)

Check if it matches (3π - 2)/(π + 6)

expected_slope = (3*math.pi - 2) / (math.pi + 6) print("Slope at t=1/4:", slope_at_1_4) print("Numerical slope:", slope_num) print("Expected slope:", expected_slope) print("Match:", math.isclose(slope_num, expected_slope, rel_tol=1e-9))

Also verify the point lies on the line

x_val = float(x.subs(t, t_val)) y_val = float(y.subs(t_val)) line_y = 3x_val - 5math.e print("x at t=1/4:", x_val) print("y at t=1/4:", y_val) print("Line y at x:", line_y) print("Match on line:", math.isclose(y_val, line_y, rel_tol=1e-9))

Extract a, b, c from the exact slope expression

slope_at_1_4 should be (3*pi - 2)/(pi + 6)

numerator = slope_at_1_4.as_numer_denom()[0] denominator = slope_at_1_4.as_numer_denom()[1]

Check if numerator is of form a*pi - b and denominator is pi + c

We'll solve for a, b, c

Assume slope = (a*pi - b)/(pi + c)

Cross multiply: (a*pi - b) = slope * (pi + c)

But easier: from our derivation, we expect a=3, b=2, c=6

Let's verify by comparing

a = 3 b = 2 c = 6 expected_form = (a*pi - b) / (pi + c) print("Expected form:", expected_form) print("Same as slope?", simplify(slope_at_1_4 - expected_form) == 0) print("a+b+c:", a+b+c) [/PYTHON]

[THOUGHT] The Python code confirms the slope at t=1/4 is (3*pi - 2)/(pi + 6), so a=3, b=2, c=6. Thus, a+b+c=11.

[/THOUGHT] \boxed{11}