JEE Math Practice Question

Let the maximum area of the triangle that can be inscribed in the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2$$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be $$6\sqrt 3 $$. Then the eccentricity of the ellipse is :

1. A.$${{\sqrt 3 } \over 2}$$
2. B.$${1 \over 2}$$
3. C.$${1 \over {\sqrt 2 }}$$
4. D.$${{\sqrt 3 } \over 4}$$