Question

If the minor axis of an ellipse subtends an equilateral triangle with vertex at one end of major axis, then write the eccentricity of the ellipse.

Answer 1

$$\begin{gathered} \text{According to the question, the minor axis of the ellipse subtends an equilateral triangle with the vertex at one end of the major axis.} \\ \text{AB=}\sqrt{a^2+b^2} \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{ABC}\mathrm{is}\mathrm{an}\mathrm{equilateral}\mathrm{triangle}. \\ \therefore \mathrm{AB}=\mathrm{BB}\text{'} \\ \Rightarrow \sqrt{a^2+b^2}=2b \\ \text{On squaring both sides, we have:} \\ {a}^2+b^2=4b^2 \\ \Rightarrow a^2=3b^2 \\ \mathrm{Now},e=\sqrt{1-\frac{b^2}{a^2}} \\ \Rightarrow e=\sqrt{1-\frac{b^2}{3b^2}} \\ \Rightarrow e=\sqrt{1-\frac{1}{3}} \\ \Rightarrow e=\sqrt{\frac{2}{3}} \end{gathered}$$