Let a tangent to the curve $$9x^2 + 16y^2 = 144$$ intersect the coordinate axes at the points $$A$$ and $$B$$. Then, the minimum length of the line segment $$AB$$ is _____

Ellipse $$x^2/16 + y^2/9 = 1$$. Tangent intercepts: $$A = (4\sec\theta, 0)$$, $$B = (0, 3\csc\theta)$$.

$$AB^2 = 16\sec^2\theta + 9\csc^2\theta = 25 + 16\tan^2\theta + 9\cot^2\theta$$.

By AM-GM: $$16\tan^2\theta + 9\cot^2\theta \geq 2\sqrt{144} = 24$$.

$$AB_{min}^2 = 49$$. $$AB_{min} = 7$$.