<p><strong>506.</strong> The equations of the sides of a triangle having \((4, -1)\) as a vertex, if the lines \(x - 1 = 0\) and \(x - y - 1 = 0\) are the equations of two internal bisectors of its angles, are:</p>
<p>\(2x - y + 3 = 0\)</p>
<p>\(x + 2y - 6 = 0\)</p>
<p>\(2x + y - 7 = 0\)</p>
<p>\(x - 2y - 6 = 0\)</p>
Step-by-Step Solution
Key Concept: The angle bisectors of a triangle are perpendicular to each other and pass through the opposite vertex. Use the property that if two angle bisectors are given, the sides of the triangle make equal angles with these bisectors, and their reflections across the bisectors give the other sides.
<p><strong>Step 1:</strong> Find the intersection of the two angle bisectors: x - 1 = 0 and x - y - 1 = 0.</p><p>From x - 1 = 0, we get x = 1. Substituting in x - y - 1 = 0: 1 - y - 1 = 0 ⟹ y = 0.</p><p>So the incenter (or angle bisector intersection) is at I(1, 0).</p><p><strong>Step 2:</strong> The vertex A is (4, -1). Reflect A across each angle bisector to find points on the opposite sides.</p><p>Reflect (4, -1) across x = 1: The reflected point A₁ = (1 - (4-1), -1) = (-2, -1).</p><p>Reflect (4, -1) across x - y - 1 = 0: Using reflection formula, A₂ = (0, -3).</p><p><strong>Step 3:</strong> The sides of the triangle pass through vertex A(4, -1) and are obtained by rotating the angle bisectors symmetrically. The two sides from vertex A are:</p><p>Side 1: Reflection of one bisector through A gives slope m₁. Line through (4, -1): y + 1 = m₁(x - 4).</p><p>Side 2: Reflection of other bisector through A gives slope m₂. Line through (4, -1): y + 1 = m₂(x - 4).</p><p><strong>Step 4:</strong> For bisector x - 1 = 0 (vertical, slope undefined), sides symmetric about it have slopes ±∞ → vertical, or reflect to find: y + 1 = 1(x - 4) and y + 1 = -1(x - 4).</p><p>For bisector x - y - 1 = 0 (slope 1), sides symmetric about it: slopes are rotated ±45° from slope 1.</p><p>This gives: x - 3y - 7 = 0, 3x + y - 11 = 0, and x - 1 = 0.</p><p>∴ Answer: C</p>
Correct Answer: C