[THOUGHT] I need to solve for a given the condition PF * PF' = 8, where P is the intersection of the tangent at (a, √2 a) with the y-axis, and F, F' are the foci of the hyperbola x²/a² - y²/a² = -1.

Step 1: Identify the hyperbola and its properties. The equation x²/a² - y²/a² = -1 can be rewritten as y²/a² - x²/a² = 1, which is a hyperbola with transverse axis along the y-axis. For hyperbola y²/A² - x²/B² = 1, foci are at (0, ±c) where c² = A² + B². Here A² = a², B² = a², so c² = a² + a² = 2a², thus c = a√2. Foci: F = (0, a√2), F' = (0, -a√2).

Step 2: Find the tangent line at point (a, √2 a). For hyperbola y²/A² - x²/B² = 1, tangent at (x₀, y₀) is (yy₀)/A² - (xx₀)/B² = 1. With A² = B² = a² and (x₀, y₀) = (a, √2 a): yy₀/a² - xx₀/a² = 1 → y(√2 a) - x(a) = a² → a√2 y - a x = a². Divide by a (a > 0): √2 y - x = a. Set x = 0 to find intersection with y-axis: √2 y = a → y = a/√2. Thus P = (0, a/√2).

Step 3: Compute distances PF and PF'. PF = |y_F - y_P| = |a√2 - a/√2| = a|√2 - 1/√2| = a| (2-1)/√2 | = a/√2. PF' = |y_P - y_F'| = |a/√2 - (-a√2)| = |a/√2 + a√2| = a|1/√2 + √2| = a| (1+2)/√2 | = 3a/√2.

Step 4: Compute product PF * PF'. PF * PF' = (a/√2) * (3a/√2) = 3a²/2.

Step 5: Set product equal to 8 and solve for a². 3a²/2 = 8 → a² = 16/3.

Step 6: Verify using Python to ensure correctness. [/THOUGHT] [PYTHON] import sympy as sp

Define symbol

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

Foci coordinates

c = sp.sqrt(2)*a F = (0, c) F_prime = (0, -c)

Point P

P = (0, a/sp.sqrt(2))

Compute distances

PF = sp.Abs(F[1] - P[1]) PF_prime = sp.Abs(P[1] - F_prime[1])

Product of distances

product = PF * PF_prime

Simplify product

product_simplified = sp.simplify(product) print("Product expression:", product_simplified)

Solve equation product = 8

equation = sp.Eq(product_simplified, 8) solution = sp.solve(equation, a**2) print("Solution for a^2:", solution)

Extract p and q from solution

p, q = solution[0].as_numer_denom() print("p + q:", p + q) [/PYTHON]

[THOUGHT] The Python output confirms that a² = 16/3, so p = 16 and q = 3. Therefore, p + q = 19. [/THOUGHT] \boxed{19}