<p>Given equation of parabola is \(y^2 = 16x\). Find the length of the focal chord \(AB\) where \(A\) is the point \((1, 4)\).</p>
Step-by-Step Solution
Key Concept: For a parabola y² = 4ax, any focal chord through point (x₁, y₁) has length = x₁ + x₂ + 2a, where x₁ and x₂ are x-coordinates of endpoints. Alternatively, use the focal chord length formula: if one endpoint is at parameter t₁, the length is 4a(t₁ + 1/t₁)².
<p><strong>Step 1:</strong> Identify parabola parameters. Given y² = 16x, we have 4a = 16, so a = 4. Focus is at F(4, 0).</p><p><strong>Step 2:</strong> Verify A(1, 4) lies on parabola: 4² = 16, 16(1) = 16 ✓</p><p><strong>Step 3:</strong> For a focal chord with one endpoint at (x₁, y₁), use the focal chord property. If A = (1, 4) is one endpoint, find the other endpoint B on the focal chord through F(4, 0).</p><p><strong>Step 4:</strong> The line through F(4, 0) and A(1, 4) has slope m = (4-0)/(1-4) = -4/3. Equation: y = -4/3(x - 4).</p><p><strong>Step 5:</strong> Substitute into y² = 16x: [-4/3(x-4)]² = 16x → 16/9(x-4)² = 16x → (x-4)² = 9x → x² - 8x + 16 = 9x → x² - 17x + 16 = 0.</p><p><strong>Step 6:</strong> Solutions: x = 1 or x = 16. So x₁ = 1, x₂ = 16. Using focal chord length formula: |AB| = x₁ + x₂ + 2a = 1 + 16 + 2(4) = 1 + 16 + 8 = 25.</p><p>∴ Answer: 25</p>
Correct Answer: 25