Hyperbola
Eccentricity of Hyperbola
Grade None
Question:
<p>The eccentricity of the hyperbola, whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is</p>
<p>\(\sqrt{3}\)</p>
<p>\(\dfrac{4}{3}\)</p>
<p>\(\dfrac{4}{\sqrt{3}}\)</p>
<p>\(\dfrac{2}{\sqrt{3}}\)</p>
Step-by-Step Solution
Key Concept: Use the relationships: latus rectum = 2b²/a, conjugate axis = 2b, and distance between foci = 2c, where c² = a² + b² for a hyperbola. These three conditions provide enough constraints to solve for eccentricity e = c/a.
<p><strong>Step 1:</strong> Write the given conditions:</p><ul><li>Latus rectum: 2b²/a = 8 → b² = 4a</li><li>Conjugate axis: 2b = (1/2) × (distance between foci)</li><li>Distance between foci: 2c</li></ul><p><strong>Step 2:</strong> From condition 2: 2b = (1/2)(2c) → 2b = c → c = 2b</p><p><strong>Step 3:</strong> Use the hyperbola relation c² = a² + b²:</p><p>(2b)² = a² + b² → 4b² = a² + b² → 3b² = a²</p><p><strong>Step 4:</strong> Substitute b² = 4a into 3b² = a²:</p><p>3(4a) = a² → 12a = a² → a = 12 (taking a ≠ 0)</p><p>Therefore: b² = 4(12) = 48, so b = 4√3</p><p><strong>Step 5:</strong> Find c: c = 2b = 2(4√3) = 8√3</p><p><strong>Step 6:</strong> Calculate eccentricity:</p><p>e = c/a = 8√3/12 = 2√3/3</p><p>∴ Answer: D</p>
Correct Answer: D