<p>The equation of a hyperbola is \(16x^2 - 9y^2 = 144\). If one of its directrices is \(5x + 9 = 0\), find the corresponding focus.</p>
<p>(a) \(\left(-\frac{5}{3}, 0\right)\)</p>
<p>(b) \((-5, 0)\)</p>
<p>(c) \(\left(\frac{5}{3}, 0\right)\)</p>
<p>(d) \((5, 0)\)</p>
Step-by-Step Solution
Key Concept: For a hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the focus and directrix are related by the property that distance from a point to focus divided by distance to directrix equals eccentricity.
<p><strong>Step 1:</strong> Convert the equation to standard form: \(\frac{x^2}{9} - \frac{y^2}{16} = 1\)</p><p><strong>Step 2:</strong> Here \(a^2 = 9\), so \(a = 3\); \(b^2 = 16\)</p><p><strong>Step 3:</strong> Find eccentricity: \(e^2 = 1 + \frac{b^2}{a^2} = 1 + \frac{16}{9} = \frac{25}{9}\), so \(e = \frac{5}{3}\)</p><p><strong>Step 4:</strong> Given directrix: \(x = -\frac{9}{5}\)</p><p><strong>Step 5:</strong> For a hyperbola, the focus is at \((ae, 0) = \left(3 \cdot \frac{5}{3}, 0\right) = (5, 0)\) for the right branch, and \((-ae, 0) = (-5, 0)\) for the left branch.</p><p><strong>Step 6:</strong> Since the directrix is at \(x = -\frac{9}{5}\), the corresponding focus on the left is \((-5, 0)\).</p><p>∴ Answer is (b).</p>
Correct Answer: b