Step-by-Step Solution
Key Concept: The tangent from an external point to a parabola is found using the slope form and applying the point condition.
Equation of tangent in slope form is $y = mx + \frac{d}{m}$. Let the tangent passes through $P(h, k)$. For $C_1$: $\frac{dy}{dx}|_{(p,q)} = \frac{2a}{q}$ and slope form gives the tangent equation. Solving the slope and intercept conditions yields the equation $3y^2 = 32x$.
Correct Answer: 32