Question

If the major axis of an ellipse is three times the minor axis, then its eccentricity is equal to

• A. $\frac{1}{3}$
  
• B. $\frac{1}{\sqrt{3}}$
  
• C. $\frac{1}{\sqrt{2}}$
  
• D. $\frac{2\sqrt{2}}{3}$

Answer 1

The correct option is D $\frac{2\sqrt{2}}{3}$

Length of the major axis =2b
Length of the minor axis =2a
According to the question, the major axis of the ellipse is three times the minor axis
i.e. 2b=3(2a)
$\Rightarrow 2b=6a$
$\Rightarrow a=2b/6$
$\Rightarrow a=b/3,b=3a$
Here, a<b, so the major and the minor axes of the ellipse are along the x-axis and y-axis, respectively.
Now, $e=\sqrt{1-\frac{a^2}{b^2}}$
$\Rightarrow e=\sqrt{1-\frac{\frac{b^2}{9}}{b^2}}$
$\Rightarrow e=\sqrt{1-\frac{1}{9}}$
$\Rightarrow e=\sqrt{\frac{8}{9}}$
$\Rightarrow e=\frac{2\sqrt{2}}{3}$