If the minimum area of the triangle formed by a tangent to the...

Equation of tangent at parametric point

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

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

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

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

$⇒ k = 2$