<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: For a parabola y² = 4ax, if one end of a focal chord is at point (x₁, y₁), the other end is at (a²/x₁, -4a²/y₁), and the focal chord length equals x₁ + x₂ + 2a (or use the direct formula: length = x₁ + x₂ + 2a where a is the semi-latus rectum parameter).
<p><strong>Step 1:</strong> Identify parabola parameters. For y² = 16x, we have 4a = 16, so a = 4. Focus is at F(4, 0).</p><p><strong>Step 2:</strong> Verify that (1, 4) lies on the parabola: (4)² = 16(1) ✓ Yes, 16 = 16.</p><p><strong>Step 3:</strong> Use the focal chord property. If one end is (x₁, y₁) = (1, 4), find the other end (x₂, y₂). For a focal chord: x₁x₂ = a² and y₁y₂ = -4a².</p><p><strong>Step 4:</strong> Calculate: x₁x₂ = 4² = 16, so x₂ = 16/1 = 16. Also, y₁y₂ = -4(16) = -64, so y₂ = -64/4 = -16.</p><p><strong>Step 5:</strong> Verify second point (16, -16) is on parabola: (-16)² = 16(16) ✓ Yes, 256 = 256.</p><p><strong>Step 6:</strong> Calculate focal chord length using the formula: Length = x₁ + x₂ + 2a = 1 + 16 + 2(4) = 1 + 16 + 8 = 25.</p><p><strong>Alternative:</strong> Distance formula: √[(16-1)² + (-16-4)²] = √[225 + 400] = √625 = 25.</p><p>∴ <strong>Answer: 25</strong></p>
Correct Answer: 25