Question

The eccentricity of the ellipse whose major axis is three times the minor axis is:

• A. \( \sqrt{2}/3 \)
• B. \( \sqrt{3}/2 \)
• C. \( 2\sqrt{2}/3 \)
• D. \( 2/\sqrt{3} \)

Hint:

To complete this type of problem even faster, remember that the eccentricity formula can be written explicitly using the axis ratio \(k = \frac{\text{Major}}{\text{Minor}} = \frac{a}{b}\). The shortcut formula is \( e = \frac{\sqrt{k^2 - 1}}{k} \). Substituting \(k = 3\) gives \( e = \frac{\sqrt{3^2 - 1}}{3} = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3}\).

Answer 1

The Correct Option is C

Concept: For a standard horizontal ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (where \(a > b\)), the geometric parameters are defined as:
• Length of the major axis \(= 2a\)
• Length of the minor axis \(= 2b\)
• The eccentricity (\(e\)) measures the elongation of the ellipse and is related to its semi-axes by the relationship: \[ b^2 = a^2(1 - e^2) \quad \implies \quad e = \sqrt{1 - \frac{b^2}{a^2}} \]

Step 1: Establishing the relationship between the semi-axes from the given condition.
The problem states that the major axis is three times the length of the minor axis: \[ \text{Major axis} = 3 \times \text{Minor axis} \] \[ 2a = 3 \times (2b) \] Dividing both sides by 2 gives: \[ a = 3b \quad \implies \quad \frac{b}{a} = \frac{1}{3} \]

Step 2: Calculating the eccentricity ($e$).
Squaring the semi-axis ratio yields: \[ \frac{b^2}{a^2} = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \] Substitute this value directly into the standard eccentricity formula: \[ e = \sqrt{1 - \frac{b^2}{a^2}} \] \[ e = \sqrt{1 - \frac{1}{9}} = \sqrt{\frac{9 - 1}{9}} = \sqrt{\frac{8}{9}} \] Simplifying the radical values separately for the numerator and the denominator: \[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \] \[ \sqrt{9} = 3 \] Thus, we find the final value for the eccentricity: \[ e = \frac{2\sqrt{2}}{3} \]