[THOUGHT] We need to find the value of (a) for the given hyperbola and the condition (\overline{PF} \times \overline{PF'} = 8). The hyperbola is (\dfrac{x^2}{a^2} - \dfrac{y^2}{a^2} = -1), with foci (F) and (F'). The point (P) is where the tangent at ((a, \sqrt{2}a)) meets the y-axis. We will compute the distances (\overline{PF}) and (\overline{PF'}) in terms of (a), set their product to 8, and solve for (a). Then, express (a^2) as (\dfrac{p}{q}) and compute (p + q).

First, the foci are at ((0, a\sqrt{2})) and ((0, -a\sqrt{2})). The point (P) is at ((0, \frac{\sqrt{2}}{2}a)). The distance (\overline{PF}) is (\left| a\sqrt{2} - \frac{\sqrt{2}}{2}a \right| = \frac{\sqrt{2}}{2}a) (since (a > 0)). The distance (\overline{PF'}) is (\left| -a\sqrt{2} - \frac{\sqrt{2}}{2}a \right| = \frac{3\sqrt{2}}{2}a). The product is (\left( \frac{\sqrt{2}}{2}a \right) \times \left( \frac{3\sqrt{2}}{2}a \right) = \frac{3}{2}a^2). Setting this equal to 8 gives (\frac{3}{2}a^2 = 8), so (a^2 = \frac{16}{3}). Thus, (p = 16), (q = 3), and (p + q = 19).

To verify, we will use Python to compute the product of the distances symbolically and solve for (a), then compute (a^2) and its fractional representation. [/THOUGHT] [PYTHON] from sympy import symbols, sqrt, Eq, solve from fractions import Fraction

a = symbols('a')

Define the distances

pf = a * sqrt(2) / 2 pf_prime = 3 * a * sqrt(2) / 2

Product of the distances

product = pf * pf_prime

Set up the equation

eq = Eq(product, 8)

Solve for a

solution = solve(eq, a)

Compute a squared

a_sq = solution[0] ** 2

Convert to fraction

f = Fraction(a_sq) p = f.numerator q = f.denominator

Compute p + q

result = p + q print(result) [/PYTHON]

[THOUGHT] The Python code computed (p + q = 19), which is the result. Since the problem asks for (p + q) and the code output is 19, this is the final answer.

\boxed{19}