If a and c are positive real numbers and the ellipse (x^2)/(4...

Radius $=3 a$ Length of major axis $=4 c$ Now, (Radius $) \lt (Half$ of the length of major axis) $$\begin{aligned} & 3 a < 2 c \\ & 9 a^2 < 4 c^2 \\ & 9 a c-9 a^2>9 a c-4 c² \end{aligned}$$$\\$9 a c-9 a^2-2 c^2 \gt 9 a c-6 c^2$Again$3 a \lt 2 c\Rightarrow 9 a c \lt 6 c^2\Rightarrow 9 a c-6 c^2 \lt 0$From (i) and (ii),$9 a c-9 a^2-2 c^2 \gt 0$