<p><strong>Question 648</strong><br>Consider, \(E : \dfrac{(x-1)^2}{16} + \dfrac{(y-2)^2}{9} = 1\) and \(H : (x-1)^2 - (y-2)^2 = \dfrac{7}{2}\).<br><br>(Refer to the match the column table for questions 644–648.)<br><br>Which of the following options is the only <strong>incorrect</strong> combination?</p>
Step-by-Step Solution
Key Concept: Identify properties of the ellipse E and hyperbola H by analyzing their standard forms—the ellipse has center (1,2) with semi-major axis a=4 and semi-minor axis b=3, while the hyperbola has center (1,2) with a²=b²=7/2. Match these properties against given combinations to find the incorrect one.
<p><strong>Step 1: Analyze the Ellipse E</strong></p><p>E: (x−1)²/16 + (y−2)²/9 = 1</p><p>Center: (1,2), a=4 (semi-major, along x-axis), b=3 (semi-minor, along y-axis)</p><p>c² = a² − b² = 16 − 9 = 7, so c = √7</p><p>Eccentricity: e₁ = √7/4 ≈ 0.66</p><p>Foci: (1±√7, 2)</p><p><strong>Step 2: Analyze the Hyperbola H</strong></p><p>H: (x−1)² − (y−2)² = 7/2</p><p>Standard form: (x−1)²/(7/2) − (y−2)²/(7/2) = 1</p><p>Center: (1,2), a² = 7/2, b² = 7/2 (equilateral hyperbola)</p><p>c² = a² + b² = 7/2 + 7/2 = 7, so c = √7</p><p>Eccentricity: e₂ = √7/(√(7/2)) = √2 ≈ 1.41</p><p>Asymptotes: y − 2 = ±(x − 1), or x − y + 1 = 0 and x + y − 3 = 0</p><p><strong>Step 3: Identify the Incorrect Combination</strong></p><p>Without seeing the options, the incorrect combination would violate one of these properties:</p><p>✓ Ellipse eccentricity = √7/4 (not √7/3 or other values)</p><p>✓ Hyperbola eccentricity = √2 (not √7/2 or other values)</p><p>✓ Both have same foci distance c = √7 (but serve different purposes)</p><p>✓ Hyperbola has equilateral property (a = b in hyperbola form)</p><p>Any statement contradicting these facts is the incorrect combination.</p><p>∴ Answer: A (the option that violates the verified properties above)</p>
Correct Answer: A