The length of the minor axis (along y-axis) of an ellipse in the standard form is ${4 \over {\sqrt 3 }}$. If this ellipse touches the line, x + 6y = 8; then its eccentricity is :

Let the equation of ellipse <br><br>${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, ($a &gt; b$) <br><br>Given 2b = ${4 \over {\sqrt 3 }}$ <br><br>$\Rightarrow$ b = ${2 \over {\sqrt 3 }}$ <br><br>We know, Equation of tangent y = mx $\pm$ $\sqrt {{a^2}{m^2} + {b^2}}$ ....(1) <br><br>Given tangent is x + 6y = 8 <br><br>$\Rightarrow$ y = $- {1 \over 6}x + {8 \over 6}$ .....(2) <br><br>By comparing (1) and (2), <br><br>m = $- {1 \over 6}$ and ${{a^2}{m^2} + {b^2}}$ = ${{16} \over 9}$ <br><br>$\Rightarrow$ ${{a^2}\left( {{1 \over {36}}} \right) + {4 \over 3}}$ = ${{16} \over 9}$ <br><br>$\Rightarrow$ ${{a^2} = 16}$ <br><br>$\therefore$ e = $\sqrt {1 - {{{b^2}} \over {{a^2}}}}$ <br><br>= $\sqrt {1 - {{{4 \over 3}} \over {16}}}$ <br><br>= $\sqrt {{{11} \over {12}}}$ = ${1 \over 2}\sqrt {{{11} \over 3}}$