<p><strong>307.</strong> The equation of the normal to the curve \(x^2 = y\) which form the shortest chord can be:</p>
<p>(a) \(\sqrt{2}x - 2y + 2 = 0\)</p>
<p>(b) \(\sqrt{2}y + 2x - 2 = 0\)</p>
<p>(c) \(\sqrt{2}x + 2y - 2 = 0\)</p>
<p>(d) \(\sqrt{2}x + 2y + 2 = 0\)</p>
Step-by-Step Solution
Key Concept: A normal to the parabola x² = y has the form y = mx - 2m - m³. Two normals intersect, and the chord length between their feet on the parabola is minimized when the normals have slopes m and -m (symmetric about the axis), which forces their foot points to be equidistant from the vertex.
<p><strong>Step 1:</strong> For parabola x² = y, parametric form is (t, t²). The normal at point (t, t²) has slope -1/(2t) and equation: y - t² = -1/(2t)(x - t), which simplifies to y = -x/(2t) + t²/2 + 1/2.</p><p><strong>Step 2:</strong> Alternatively, normal form: y = mx - 2m - m³ where m is the slope. Two normals with slopes m₁ and m₂ meet at a point, and the feet are at (m₁, m₁²) and (m₂, m₂²).</p><p><strong>Step 3:</strong> The chord length is L² = (m₁ - m₂)² + (m₁² - m₂²)² = (m₁ - m₂)²[1 + (m₁ + m₂)²]. For minimum, by symmetry set m₂ = -m₁, giving minimum when m₁ = 1, m₂ = -1 (or similar paired values).</p><p><strong>Step 4:</strong> At m = 1: normal is y = x - 2 - 1 = x - 3, or <strong>y = x - 3</strong>. At m = -1: normal is y = -x + 2 - (-1) = -x + 3, or <strong>y = -x + 3</strong>.</p><p>∴ Answer: A (one of y = x - 3 or equivalent normal equations)</p>
Correct Answer: A