<p>Let <em>ABCD</em> be a parallelogram, the equations of whose diagonals are \(AC: x + 2y - 3 = 0\) and \(BD: 2x + y - 3 = 0\). If the length of the diagonal \(AC = 4\) units and the area of the parallelogram \([ABCD] = 8\) square units. The length of side <em>BD</em> is:</p>
Step-by-Step Solution
Key Concept: In a parallelogram, diagonals bisect each other at their intersection point. The area formula relates to both diagonals and the angle between them: Area = (1/2)|d₁||d₂|sin(θ), where θ is the angle between diagonals.
<p><strong>Step 1:</strong> Find the intersection point of diagonals AC and BD by solving simultaneously:</p><p>AC: x + 2y - 3 = 0</p><p>BD: 2x + y - 3 = 0</p><p>From first equation: x = 3 - 2y</p><p>Substitute in second: 2(3 - 2y) + y - 3 = 0 ⟹ 6 - 4y + y - 3 = 0 ⟹ y = 1, x = 1</p><p>Intersection point O = (1, 1)</p><p><strong>Step 2:</strong> Find the angle between diagonals using direction vectors.</p><p>For AC: x + 2y - 3 = 0, direction vector: <strong>m₁</strong> = (2, -1) (perpendicular to normal (1, 2))</p><p>For BD: 2x + y - 3 = 0, direction vector: <strong>m₂</strong> = (1, -2) (perpendicular to normal (2, 1))</p><p>cos(θ) = |2(1) + (-1)(-2)|/[√(4+1)√(1+4)] = |2 + 2|/(√5 × √5) = 4/5</p><p>sin²(θ) = 1 - 16/25 = 9/25 ⟹ sin(θ) = 3/5</p><p><strong>Step 3:</strong> Use the area formula for a parallelogram:</p><p>Area = (1/2)|AC||BD|sin(θ)</p><p>8 = (1/2) × 4 × |BD| × (3/5)</p><p>8 = (6/5)|BD|</p><p>|BD| = 40/6 = 20/3 units</p><p>∴ Answer: A</p>
Correct Answer: A