Explanation:

- The ellipse equation is \(\frac{x^2}{9} + \frac{y^2}{16} = 1\).

- The general form of the line equation tangent to an ellipse has the condition \(c = \sqrt{a^2 \cos^2 \theta + b^2 \sin^2 \theta}\).

- Line \(x + y = a\) can be compared with the tangent condition \(c = a\).

- Substitute \(a = 3, b = 4\) (semi-major and semi-minor axes of the ellipse):

- Find \(a\) such that substitution in \((x/a)^2 + (y/b)^2\) gives 1.

- Apply \((3\cos(\theta))^2 + (4\sin(\theta))^2 = a^2\).

- Solve: \(a^2 = (\frac{3\cdot 1}{\sqrt{2}})^2 + (\frac{4\cdot 1}{\sqrt{2}})^2\).

- Simplify: \(a^2 = 9/2 + 16/2 = 25/2\).

- Recheck the value: Solving using tangent conditions, obtain \(a = \pm 5\).

Answer: \(\textcolor{green}{\text{}}\) Option: 2, \(\pm 5\)