[THOUGHT] The given hyperbola is $\dfrac{x^2}{a^2} - \dfrac{y^2}{a^2} = -1$, which can be rewritten as $\dfrac{y^2}{a^2} - \dfrac{x^2}{a^2} = 1$. This is a hyperbola centered at the origin opening upwards and downwards with $a = b$. The foci are at $(0, \pm a\sqrt{2})$.

The tangent line at $(a, \sqrt{2}a)$ is derived using the hyperbola tangent formula:

Setting $x = 0$ (intersection with the y-axis) gives $y = \frac{a}{\sqrt{2}}$. Thus, point $P$ is $(0, \frac{a}{\sqrt{2}})$.

Distances from $P$ to the foci $F(0, a\sqrt{2})$ and $F'(0, -a\sqrt{2})$ are:

Their product is:

Setting this equal to 8:

Thus, $p = 16$ and $q = 3$, giving $p + q = 19$.

[PYTHON]

Verification of the solution

from sympy import symbols, solve, sqrt

a = symbols('a', positive=True, real=True)

Equation from the product of distances

equation = (3 * a2) / 2 - 8 solution = solve(equation, a2) a_squared = solution[0] p, q = a_squared.as_numer_denom() print(p + q) [/PYTHON]

[THOUGHT] The Python output confirms that $p + q = 19$ when $a^2 = \frac{16}{3}$. This matches the derived solution. [/THOUGHT] $\boxed{19}$