Let a variable line of slope $$m > 0$$ passing through the point $$(4, -9)$$ intersect the coordinate axes at the points $$A$$ and $$B$$. The minimum value of the sum of the distances of $$A$$ and $$B$$ from the origin is

Line through (4, -9) with slope m > 0: $$y + 9 = m(x - 4)$$.

x-intercept A: $$y = 0 \Rightarrow x = 4 + 9/m$$. OA = 4 + 9/m.

y-intercept B: $$x = 0 \Rightarrow y = -9 - 4m$$. OB = |−9 − 4m| = 9 + 4m (since m > 0).

Sum = OA + OB = 4 + 9/m + 9 + 4m = 13 + 9/m + 4m.

By AM-GM: $$9/m + 4m \geq 2\sqrt{36} = 12$$. Equality when 9/m = 4m, m = 3/2.

Minimum sum = 13 + 12 = 25.

The correct answer is Option (2): 25.