[THOUGHT] We need the intersection point $P$ of curve $C$ and line $y = 3x - 5e$.
Given parametric equations:

Set $y = 3x - 5e$:

Simplify left side: $1 - 3\cos^2\pi t = 1 - \frac{3}{2}(1+\cos 2\pi t) = \frac{2 - 3 - 3\cos 2\pi t}{2} = \frac{-1 - 3\cos 2\pi t}{2}$.
But more directly, bring terms together:

Using $\cos^2 + \sin^2 = 1$, the bracket becomes $1 - 3\cos^2 - 3 - 3\sin^2 = -2 - 3(\cos^2 + \sin^2) = -5$.
Thus $-5 e^{4t} = -5e \Rightarrow e^{4t} = e \Rightarrow t = \frac{1}{4}$.

Now compute the slope $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.
First, evaluate functions at $t = 1/4$:

Derivative of $x$:

At $t = 1/4$: $\sin\pi/4 = \sqrt2/2$, $\sin^2 = 1/2$, $\sin 2\pi t = \sin(\pi/2) = 1$.
Thus

Derivative of $y$:

so $\frac{d}{dt}(-3\cos^2\pi t) = -3(-\pi\sin2\pi t) = 3\pi\sin2\pi t$.
At $t = 1/4$: $\cos^2=1/2$, so $1 - 3\cos^2 = -1/2$; $\sin2\pi t = \sin(\pi/2)=1$.
Hence

Therefore,

This matches the form $\frac{a\pi - b}{\pi + c}$ with $a = 3$, $b = 2$, $c = 6$.
Thus $a + b + c = 3 + 2 + 6 = 11$.

$\boxed{11}$

[THOUGHT] To solve the problem, we need to find the slope of the tangent to the curve $C$ at point $P$ where it intersects the line $y = 3x - 5e$. The parametric equations for $C$ are:

First, we find the value of $t$ at the intersection point by solving $y = 3x - 5e$. Substituting the parametric forms:

Simplifying leads to $e^{4t} = e$, so t = $$\frac{1}{4}$$.

Next, we compute the derivative $$\frac{dy}{dx}$$ = $$\frac{dy/dt}{dx/dt}$$ at t = $$\frac{1}{4}$$. Using symbolic differentiation:

• $$\frac{dx}{dt}$$ = e^{4t} \left[ 4(1 + \sin^2 $$\pi$$ t) + $$\pi$$ \sin 2$$\pi$$ t \right]
• $$\frac{dy}{dt}$$ = e^{4t} \left[ 4(1 - 3\cos^2 $$\pi$$ t) + 3$$\pi$$ \sin 2$$\pi$$ t \right]

At t = $$\frac{1}{4}$$, \sin($$\pi$$/4) = \cos($$\pi$$/4) = $$\frac{\sqrt{2}}{2}$$, so \sin^2($$\pi$$/4) = \cos^2($$\pi$$/4) = $$\frac{1}{2}$$, and \sin($$\pi$$/2) = 1. Substituting these values:

• $$\frac{dx}{dt}$$ = e($$\pi$$ + 6)
• $$\frac{dy}{dt}$$ = e(3$$\pi$$ - 2)

Thus, the slope is:

This matches the form $$\frac{a\pi - b}{\pi + c}$$ with $a = 3$, $b = 2$, $c = 6$. Therefore, $a + b + c = 11$.

[PYTHON] import math

Calculate the exact slope value numerically for verification

pi = math.pi t = 0.25

Compute sin(pit), cos(pit)

sin_pi_t = math.sin(pi * t) cos_pi_t = math.cos(pi * t) sin_sq = sin_pi_t ** 2 cos_sq = cos_pi_t ** 2

Compute dx/dt

exp_4t = math.exp(4 * t) dx_dt = exp_4t * (4 * (1 + sin_sq) + pi * math.sin(2 * pi * t))

Compute dy/dt

dy_dt = exp_4t * (4 * (1 - 3 * cos_sq) + 3 * pi * math.sin(2 * pi * t))

Slope dy/dx

slope = dy_dt / dx_dt

Exact expression (3π - 2)/(π + 6)

exact_slope = (3 * pi - 2) / (pi + 6)

print("Computed slope:", slope) print("Exact slope:", exact_slope) print("a=3, b=2, c=6, sum =", 3 + 2 + 6) [/PYTHON]

The output confirms the slope is indeed $$\frac{3\pi - 2}{\pi + 6}$$, so $a = 3$, $b = 2$, $c = 6$, and their sum is $11$.

$\boxed{11}$

[THOUGHT] The Python output confirms that the slope is $\frac{3\pi - 2}{\pi + 6}$, so $a = 3$, $b = 2$, $c = 6$, and $a + b + c = 11$. Therefore, the final answer is:

$\boxed{11}$