<p>If one end of a focal chord of the parabola \(y^2 = 16x\) is at \((1, 4)\), then the length of this focal chord is</p>
Step-by-Step Solution
Key Concept: Use the focal chord length formula \(4a(1 + \cot^2 \alpha)\) where \(\alpha\) is the inclination angle of the chord with the x-axis.
<p><strong>Step 1:</strong> The parabola is \(y^2 = 16x\), so \(4a = 16\) which gives \(a = 4\). The focus is at \((4, 0)\).</p><p><strong>Step 2:</strong> Find the slope of the focal chord with one endpoint at \((1, 4)\). The slope is \(m = \tan \alpha = \frac{4 - 0}{1 - 4} = -\frac{4}{3}\).</p><p><strong>Step 3:</strong> Thus \(\cot \alpha = -\frac{3}{4}\), so \(\cot^2 \alpha = \frac{9}{16}\).</p><p><strong>Step 4:</strong> The length of a focal chord is \(4a(1 + \cot^2 \alpha) = 16\left(1 + \frac{9}{16}\right) = 16 \cdot \frac{25}{16} = 25\) units.</p><p>∴ Answer is (b) 25.</p>
Correct Answer: B