If the tangent at a point on the ellipse x ^ 2 27 + y ^ 2 3 = 1...

$\dfrac{{x}^{2}}{27} + \dfrac{{y}^{2}}{3} = 1$

Equation of tangent to ellipse

$\dfrac{x}{\sqrt{27}}cos\theta + \dfrac{y}{\sqrt{3}}sin\theta = 1$

Area bounded by the line and coordinate axis

$\Delta = \dfrac{1}{2} \cdot \dfrac{\sqrt{27}}{cos\theta} \cdot \dfrac{\sqrt{3}}{sin\theta} = \dfrac{9}{\sin 2\theta}$

$\Delta$ will be minimum when $\sin 2\theta = 1$

{\Delta}_\min = 9 sq. units