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

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