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)

Step 1: Understanding the Concept:
This problem relates the lengths of the major axis (2a), minor axis (2b), and the eccentricity (e) of an ellipse. We need to use the standard formulas that connect these three properties.
Step 2: Key Formula or Approach:
1. Length of major axis, \(m = 2a\). 2. Length of minor axis, \(n = 2b\). 3. The relationship between a, b, and eccentricity e is \(b^2 = a^2(1 - e^2)\), which can be rearranged to \(a^2 - b^2 = a^2e^2\).
Step 3: Detailed Explanation:
We are given:
\(m^2 - n^2 = 45\)
\(e = \frac{\sqrt{5}}{3}\)
Substitute \(m = 2a\) and \(n = 2b\) into the first equation: \[ (2a)^2 - (2b)^2 = 45 \] \[ 4a^2 - 4b^2 = 45 \] \[ 4(a^2 - b^2) = 45 \] Now, use the identity \(a^2 - b^2 = a^2e^2\): \[ 4(a^2e^2) = 45 \] We are given \(e = \frac{\sqrt{5}}{3}\), so \(e^2 = \left(\frac{\sqrt{5}}{3}\right)^2 = \frac{5}{9}\). Substitute this value into the equation: \[ 4\left(a^2 \cdot \frac{5}{9}\right) = 45 \] \[ a^2 \cdot \frac{20}{9} = 45 \] Solve for \(a^2\): \[ a^2 = 45 \cdot \frac{9}{20} = \frac{9 \cdot 5 \cdot 9}{4 \cdot 5} = \frac{81}{4} \] Take the square root to find `a`: \[ a = \sqrt{\frac{81}{4}} = \frac{9}{2} \] The question asks for the length of the major axis, which is \(m = 2a\). \[ m = 2 \cdot \frac{9}{2} = 9 \] Step 4: Final Answer:
The length of the major axis is 9.