Question

If the length of the major axis of an ellipse is 3 times the length of the minor axis, then its eccentricity is:

• A. \( \frac{1}{\sqrt{2}} \)
• B. \( \frac{2\sqrt{5}}{3} \)
• C. \( \frac{2}{\sqrt{3}} \)
• D. \( \frac{1}{\sqrt{3}} \)

Hint:

When solving for the eccentricity of an ellipse, remember that the relationship between the semi-major axis \( a \) and semi-minor axis \( b \) is key. In this case, we used the fact that \( a = 3b \).

Answer 1

The Correct Option is B

For an ellipse, the eccentricity \( e \) is given by: \[ e = \sqrt{1 - \frac{b^2}{a^2}} \] where \( a \) is the semi-major axis and \( b \) is the semi-minor axis. We are told that the length of the major axis is 3 times the length of the minor axis. Therefore: \[ a = 3b \] Substitute this into the formula for eccentricity: \[ e = \sqrt{1 - \frac{b^2}{(3b)^2}} = \sqrt{1 - \frac{b^2}{9b^2}} = \sqrt{1 - \frac{1}{9}} = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3} \] Thus, the correct answer is option (B)