Question

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

• A. $\frac{1}{2}\sqrt{\frac{5}{3}}$
• B. $\frac{1}{2}\sqrt{\frac{11}{3}}$
• C. $\sqrt{\frac{5}{6}}$
• D. $\frac{1}{3}\sqrt{\frac{11}{3}}$

Answer 1

The correct option is B $\frac{1}{2}\sqrt{\frac{11}{3}}$

If $2b=\frac{4}{\sqrt{3}}$
$b=\frac{2}{\sqrt{3}}$
Comparing $y=-\frac{x}{6}+\frac{8}{6}$ with $y=mx\pm \sqrt{a^2{m}^2+b^2}$
$m=-\frac{1}{6}$ and $a^2{m}^2+b^2=\frac{16}{9}$
$\frac{a^2}{36}+\frac{4}{3}=\frac{16}{9}$
$\Rightarrow \frac{a^2}{36}=\frac{16}{9}-\frac{4}{3}$
$\Rightarrow a^2=16$
$e=\sqrt{1-\frac{b^2}{a^2}}$
$\Rightarrow e=\sqrt{\frac{11}{12}}$