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

Let the equation of ellipse is
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ where a>b
The minor axis of an ellipse subtends an equilateral triangle with vertex at one end of major axis.
$\therefore △ABC$ is an equilateral
$\therefore AB=BC=CA$
Now,
AB=AC
$\Rightarrow A{B}^2=A{C}^2$
$\Rightarrow (0-0)+(b+b{)}^2=(0-a{)}^2+(b-0{)}^2$
$\Rightarrow (2b{)}^2+(-a{)}^2+(b{)}^2$
$\Rightarrow 4b^2=a^2+b^2$
$\Rightarrow 3b^2=a^2$
$\Rightarrow b^2=\frac{a^2}{3}$ ...(i)
Now,
$\Rightarrow b^2=a^2(1-e^2)$
$\Rightarrow \frac{a^2}{3}=a^2(1-e^2)$[Using equation (i)]
$\Rightarrow \frac{1}{3}=1-e^2$
$e^2=1-\frac{1}{3}$
$e^2=\frac{2}{3}$
$e=\sqrt{\frac{2}{3}}$