Question

If S and S' are two foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and B is an end of the minor axis such that ∆BSS' is equilateral, then write the eccentricity of the ellipse.

Answer 1

$$\begin{gathered} We\mathrm{know}\mathrm{that}the\mathrm{focal}\mathrm{distance}\mathrm{of}\mathrm{a}\mathrm{point}B(0,b)\mathrm{is}a\pm e.0=a \\ \mathrm{i}.\mathrm{e}.SB=SB\text{'}=a \\ \therefore SB+S{B}^{\text{'}}=2a \\ \mathrm{Since}∆{\text{BSS}}^{\text{'}}\text{is}\text{equilateral, we have:} \\ SB=S{S}^{\text{'}}=S^{\text{'}}B=2ae \\ \Rightarrow 2ae+2ae=2a \\ \Rightarrow 4ae=2a \\ \Rightarrow e=\frac{2}{4} \\ \Rightarrow e=\frac{1}{2} \end{gathered}$$