Question

If eccentricity \(e = \frac13\), then the ratio of the major axis to the minor axis of the ellipse is:

• A. \(3 : 2\sqrt2\)
• B. \(9 : 8\)
• C. \(2\sqrt2 : 3\)
• D. \(8 : 9\)

Hint:

For ellipse ratios, start with \(e = \fracca\) and use \(b^2 = a^2 - c^2\) to relate axes lengths.

Answer 1

The Correct Option is A

For an ellipse, the eccentricity is given by:
\[ e = \fracca \] where \(a\) is the semi-major axis and \(c\) is the distance from the center to the focus.
We are given \(e = \frac13\), so:
\[ \fracca = \frac13 \Rightarrow c = \fraca3 \] We know that for an ellipse:
\[ b^2 = a^2 - c^2 \] Substitute \(c = \fraca3\):
\[ b^2 = a^2 - \fraca^29 = a^2\left(1 - \frac19\right) = a^2 \frac89 \] Thus:
\[ b = \frac2\sqrt23a \] The major axis length = \(2a\), and the minor axis length = \(2b = 2 \frac2\sqrt23a = \frac4\sqrt23a\).
The ratio of major to minor axis:
\[ \frac2a\frac4\sqrt23a = \frac2\frac4\sqrt23 = \frac2 34\sqrt2 = \frac32\sqrt2 \] So the ratio is \(3 : 2\sqrt2\).