Equation of tangent at parametric point 

 $\left(b \cos \theta ,2a \sin \theta \right)$ to the ellipse is $\frac{x}{b}\cos \theta +\frac{y}{2a}\sin \theta =1$

Co-ordinates of $P\left(\frac{b}{\cos \theta },0\right)$ and $Q\left(0,\frac{2a}{\sin \theta }\right)$ respectively.

Therefore, Area of triangle formed by tangent and co-ordinate axes$=\frac{1}{2}\times \left|\frac{b}{\cos \theta }\times \frac{2a}{\sin \theta }\right|=\frac{ab}{\sin \theta .\cos \theta }=\frac{2ab}{\sin 2\theta }$

as $\sin \theta \le 1$ therefore we can say Minimum area of triangle $=2ab$

$\Rightarrow k=2$