Question:

<p><strong>Step 1:</strong> Given parabola: \(x^2 = 4y\) and chord: \(x - 2y + 4\sqrt{2} = 0\)</p><p><strong>Step 2:</strong> From the chord equation: \(x = 2y - 4\sqrt{2}\)</p><p><strong>Step 3:</strong> Substitute into parabola equation:</p><p>\[2(y - 4\sqrt{2})^2 = 4y\]</p><p>\[(y - 4\sqrt{2})^2 = 2y\]</p><p>\[y^2 - 8\sqrt{2}y + 32 = 2y\]</p><p>\[y^2 - 10y + 16 = 0\]</p><p><strong>Step 4:</strong> By Vieta's formulas: \(y_1 + y_2 = 10\) and \(y_1 y_2 = 16\)</p><p><strong>Step 5:</strong> From \(x = 2y - 4\sqrt{2}\), we get \(x^2 - 2\sqrt{2}x - 16 = 0\)</p><p>Thus \(x_1 + x_2 = 2\sqrt{2}\) and \(x_1 x_2 = -16\)</p><p><strong>Step 6:</strong> Length of chord:</p><p>\[AB = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\]</p><p>\[= \sqrt{(x_1+x_2)^2 - 4x_1x_2 + (y_1+y_2)^2 - 4y_1y_2}\]</p><p>\[= \sqrt{8 + 64 + 100 - 64} = \sqrt{108} = 6\sqrt{3}\]</p><p>∴ Answer is (d).</p>