Question

If the major axis of an ellipse is three times the minor axis, then its eccentricity is equal to
(a) $\frac{1}{3}$

(b) $\frac{1}{\sqrt{3}}$

(c) $\frac{1}{\sqrt{2}}$

(d) $\frac{2\sqrt{2}}{3}$

(e) $\frac{2}{3\sqrt{2}}$

Answer 1

$$\begin{gathered} \left(\mathrm{d}\right)\frac{2\sqrt{2}}{3} \\ \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{major}\mathrm{axis}=2b \\ \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{minor}\mathrm{axis}=2a \\ \mathrm{According}\mathrm{to}\mathrm{question},\mathrm{the}\mathrm{major}\mathrm{axis}\mathrm{of}\mathrm{the}\mathrm{ellipse}\mathrm{is}\mathrm{three}\mathrm{times}\mathrm{the}\mathrm{minor}\mathrm{axis}. \\ \mathrm{i}.\mathrm{e}.2b=3\left(2a\right) \\ \Rightarrow 2b=6a \\ \Rightarrow a=2b/6 \\ \Rightarrow a=b/3,b=3a \\ \mathrm{Here},a<b,s\mathrm{o}\mathrm{the}\mathrm{major}\mathrm{and}\mathrm{the}\mathrm{minor}\mathrm{axes}\mathrm{of}\mathrm{the}\mathrm{ellipse}\mathrm{are}\mathrm{along}\mathrm{the}x-\mathrm{axis}\mathrm{and}\mathrm{the}y-\mathrm{axis},\mathrm{respectively}. \\ \mathrm{Now},e=\sqrt{1-\frac{a^{{}^2}}{b^2}} \\ \Rightarrow e=\sqrt{1-\frac{{\displaystyle \frac{b^2}{9}}}{b^2}} \\ \Rightarrow e=\sqrt{1-\frac{1}{9}} \\ \Rightarrow e=\sqrt{\frac{8}{9}} \\ \Rightarrow e=\frac{2\sqrt{2}}{3} \end{gathered}$$