Equation of tangent to given ellipse is
$y= mx +\sqrt {a^2m^2+b^2}$
It meets axes at $\bigg(-\large\frac{\sqrt {a^2m^2 +b^2}}{m},$$0 \bigg)$ and $ (0, \sqrt {a^2m^2+b^2})$
$\therefore $ Area of triangle
$\qquad= \large\frac{1}{2} \bigg| \large\frac{a^2m^2 +b^2}{m} \bigg|$
$\qquad= \large\frac{1}{2} $$ \bigg| a^2m +\large\frac{b^2}{m} \bigg| $$ \geq ab$
Hence A is the correct answer.