<p>If \(y = x\) and \(3y + 2x = 0\) are the equations of a pair of conjugate diameters of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) then the eccentricity is ___.</p>
Step-by-Step Solution
Key Concept: For an ellipse, if two lines are conjugate diameters, their slopes m₁ and m₂ satisfy: m₁·m₂ = -b²/a². Use this property with the given slopes to find the relationship between a and b, then calculate eccentricity.
<p><strong>Step 1:</strong> Identify the slopes of the two lines.</p><p>From y = x: slope m₁ = 1</p><p>From 3y + 2x = 0: y = -⅔x, so slope m₂ = -⅔</p><p><strong>Step 2:</strong> Apply the conjugate diameter condition.</p><p>For conjugate diameters of ellipse x²/a² + y²/b² = 1:</p><p>m₁ · m₂ = -b²/a²</p><p><strong>Step 3:</strong> Substitute and solve.</p><p>(1)·(-⅔) = -b²/a²</p><p>-⅔ = -b²/a²</p><p>b²/a² = ⅔</p><p>b² = ⅔a²</p><p><strong>Step 4:</strong> Calculate eccentricity.</p><p>e² = 1 - b²/a² = 1 - ⅔ = ⅓</p><p>e = 1/√3 = √3/3</p><p>∴ Answer: <strong>1/√3 or √3/3</strong></p>
Correct Answer: 1