<p>A square of side \(a\) lies above the \(x\)-axis and has one vertex at the origin. The side passing through the origin makes an angle \(\alpha\left(0 < \alpha < \dfrac{\pi}{4}\right)\) with the positive direction of \(x\)-axis. The equation of its diagonal not passing through the origin is:</p>
<p>\(y(\cos\alpha - \sin\alpha) - x(\sin\alpha - \cos\alpha) = a\)</p>
<p>\(y(\cos\alpha + \sin\alpha) + x(\sin\alpha - \cos\alpha) = a\)</p>
<p>\(y(\cos\alpha + \sin\alpha) + x(\sin\alpha + \cos\alpha) = a\)</p>
<p>\(y(\cos\alpha + \sin\alpha) + x(\cos\alpha - \sin\alpha) = a\)</p>
Step-by-Step Solution
Key Concept: The square has one vertex at origin with one side along a line making angle α with x-axis. Use rotation of coordinates: if one side goes through origin at angle α, the four vertices are at O(0,0), A(a cos α, a sin α), B(a cos α - a sin α, a sin α + a cos α), C(-a sin α, a cos α). The equation of the side opposite to the one through origin is found by translating the line through O by perpendicular distance a.
<p><strong>Step 1:</strong> The side through origin makes angle α with x-axis. This side has equation: y = x tan α, or x sin α - y cos α = 0.</p><p><strong>Step 2:</strong> The perpendicular direction to this line is (sin α, -cos α). The opposite parallel side is at perpendicular distance a away.</p><p><strong>Step 3:</strong> Moving distance a in the perpendicular direction (away from origin, into the square): the opposite side passes through point (a sin α, -a cos α) + (a sin α, -a cos α) rotated appropriately.</p><p><strong>Step 4:</strong> The opposite side has equation: x sin α - y cos α = a sin²α + a cos²α = a, which simplifies to x sin α - y cos α = a.</p><p><strong>Step 5:</strong> Alternatively, use the fact that both parallel sides satisfy x sin α - y cos α = k. For the side through origin, k = 0. For the opposite side at distance a perpendicular away: k = a.</p><p>∴ Answer: <strong>x sin α - y cos α = a</strong></p>
Correct Answer: D