A) \[\frac{1}{3}\sqrt{\frac{11}{3}}\]
B) \[\frac{1}{2}\sqrt{\frac{5}{3}}\]
C) \[\sqrt{\frac{5}{6}}\]
D) \[\frac{1}{2}\sqrt{\frac{11}{3}}\]
Correct Answer: D
Solution :
\[2b=\frac{4}{\sqrt{3}}\,\,\Rightarrow \,\,{{b}^{2}}=\frac{4}{3}\] Equation of tangent to ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[y=mx\pm \sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}},\] here \[m=-\frac{1}{6}\] so equation of tangent is \[y=-\frac{x}{6}\pm \sqrt{\frac{{{a}^{2}}}{36}+\frac{4}{3}}\] But \[x+6y=8\] is given to be a tangent So after comparing we get \[a=4\] Now \[e=\sqrt{1-\frac{{{b}^{2}}}{{{a}^{2}}}}=\sqrt{\frac{11}{12}}\]
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