<p>If \(5x + 9 = 0\) is the directrix of the hyperbola \(16x^2 - 9y^2 = 144\), then its corresponding focus is:</p>
<p>\((5, 0)\)</p>
<p>\(\left(-\dfrac{5}{3}, 0\right)\)</p>
<p>\(\left(\dfrac{5}{3}, 0\right)\)</p>
<p>\((-5, 0)\)</p>
Step-by-Step Solution
Key Concept: For a hyperbola, the eccentricity relates the directrix and focus through the formula: distance from center to directrix = a/e, and distance from center to focus = c = ae. Use the standard form to find a, b, c, and e, then locate the focus corresponding to the given directrix.
<p><strong>Step 1:</strong> Convert to standard form. Given: 16x² - 9y² = 144</p><p>Divide by 144: x²/9 - y²/16 = 1</p><p>This is a hyperbola with horizontal transverse axis where a² = 9 and b² = 16, so a = 3 and b = 4.</p><p><strong>Step 2:</strong> Find eccentricity. For hyperbola: c² = a² + b² = 9 + 16 = 25, so c = 5</p><p>Eccentricity: e = c/a = 5/3</p><p><strong>Step 3:</strong> Verify the directrix. Directrix formula for horizontal hyperbola: x = ±a/e</p><p>x = ±(3)/(5/3) = ±9/5 = ±1.8</p><p>Given directrix is 5x + 9 = 0, which gives x = -9/5 = -1.8 ✓</p><p><strong>Step 4:</strong> Find corresponding focus. For the left directrix (x = -a/e), the corresponding focus is at (-c, 0)</p><p>Focus: (-5, 0)</p><p>∴ Answer: A</p>
Correct Answer: A