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

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