If two distinct tangents can be drawn from the point $(a, 2)$ on different branches of the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$, then
Step-by-Step Solution
Key Concept: For tangents from external points on different branches, the point must lie on the asymptote between the two asymptotic directions.
For two distinct tangents on different branches, the point must lie on the asymptote line $y = 2$ between the two asymptotes $4x = \pm 3y$. Solving $y = 2$ with asymptote $4x = 3y$ gives $x = \frac{3}{2}$, so the point must satisfy $-\frac{3}{2} < a < \frac{3}{2}$ to ensure tangents touch different branches.
Correct Answer: 1