Question

The length of major axis and minor axis of an ellipse are, respectively, \( m \) and \( n \). If \( m^2 - n^2 = 45 \) and the eccentricity of the ellipse is \( \frac{\sqrt{5}}{3} \), then the length of the major axis is

• A. \(13 \)
• B. \(6 \)
• C. \(12 \)
• D. \(18 \)
• E. \(9 \)

Hint:

Convert axis lengths into \(a, b\) form before applying ellipse formulas.

Answer 1

Answer: (E)

Concept: \[ e = \frac{c}{a}, \quad c^2 = a^2 - b^2 \]

Step 1: Relate axes.
\[ m = 2a, \quad n = 2b \] \[ m^2 - n^2 = 4(a^2 - b^2) = 45 \] \[ a^2 - b^2 = \frac{45}{4} \]

Step 2: Use eccentricity.
\[ e^2 = \frac{c^2}{a^2} = \frac{a^2 - b^2}{a^2} \] \[ \frac{5}{9} = \frac{45/4}{a^2} \] \[ a^2 = \frac{45}{4} \cdot \frac{9}{5} = \frac{81}{4} \Rightarrow a = \frac{9}{2} \]

Step 3: Find major axis.
\[ m = 2a = 9 \]