<p>If \(S_1\) and \(S_2\) are the foci of the hyperbola whose transverse axis length is 4 and conjugate axis length is 6, \(S_3\) and \(S_4\) are the foci of the conjugate hyperbola, then the area of the quadrilateral \(S_1 S_3 S_2 S_4\) is \(k\). Find \(k/4\).</p>
Step-by-Step Solution
Key Concept: For a hyperbola and its conjugate hyperbola, the foci form a rectangle because they share the same relationship between a and b. The foci of the original hyperbola lie on the transverse axis while the foci of the conjugate hyperbola lie on the conjugate axis, creating perpendicular configurations.
<p><strong>Step 1: Find parameters of the original hyperbola.</strong></p><p>Transverse axis length = 2a = 4 ⟹ a = 2</p><p>Conjugate axis length = 2b = 6 ⟹ b = 3</p><p><strong>Step 2: Find c for the original hyperbola.</strong></p><p>c² = a² + b² = 4 + 9 = 13 ⟹ c = √13</p><p>Foci S₁ and S₂ are at (±√13, 0)</p><p><strong>Step 3: Find parameters of the conjugate hyperbola.</strong></p><p>For the conjugate hyperbola: a' = b = 3, b' = a = 2</p><p>c'² = a'² + b'² = 9 + 4 = 13 ⟹ c' = √13</p><p>Foci S₃ and S₄ are at (0, ±√13)</p><p><strong>Step 4: Find the area of quadrilateral S₁S₃S₂S₄.</strong></p><p>The four foci form a rectangle with:</p><p>• Vertices: S₁(√13, 0), S₃(0, √13), S₂(-√13, 0), S₄(0, -√13)</p><p>• Diagonals: S₁S₂ has length 2√13 (horizontal)</p><p>• Diagonals: S₃S₄ has length 2√13 (vertical)</p><p>This is a rhombus/square rotated 45°. Using the diagonal formula for a rhombus:</p><p>Area = (1/2) × d₁ × d₂ = (1/2) × 2√13 × 2√13 = (1/2) × 4 × 13 = 26</p><p>Therefore k = 26</p><p><strong>∴ k/4 = 26/4 = 13/2 = 6.5</strong></p><p><em>Note: If answer format expects integer, verify k/4 = 26/4; if simplified form needed, answer is 13/2 or 6.5</em></p>
Correct Answer: 13