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The length of the minor axis (along $y-$axis) of an ellipse in the standard form is $\frac{4}{\sqrt{3}} .$ If this ellipse touches the line, $x+6 y=8 ;$ then its eccentricity is
$\sqrt{\frac{5}{6}}$
$\frac{1}{2} \sqrt{\frac{11}{3}}$
$\frac{1}{3} \sqrt{\frac{11}{3}}$
$\frac{1}{2} \sqrt{\frac{5}{3}}$
Solution
Let $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{b}^{2}}=1 ; \mathrm{a}>\mathrm{b}$
$2 b=\frac{4}{\sqrt{3}} \Rightarrow b=\frac{2}{\sqrt{3}} \Rightarrow b^{2}=\frac{4}{3}$
tangent $\mathrm{y}=\frac{-\mathrm{x}}{6}+\frac{4}{3}$ compare with
$\mathrm{y}=\mathrm{mx} \pm \sqrt{\mathrm{a}^{2} \mathrm{m}^{2}+\mathrm{b}^{2}}$
$\Rightarrow \mathrm{m}=\frac{-1}{6} \Rightarrow \sqrt{\frac{\mathrm{a}^{2}}{36}+\frac{4}{3}}=\frac{4}{3} \Rightarrow \mathrm{a}=4$
$e=\sqrt{1-\frac{b^{2}}{a^{2}}}=\frac{1}{2} \sqrt{\frac{11}{3}}$
