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

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