Area Under the Curve
Area bounded by modulus inequalities
Grade 12
Question:
<p>If area bounded by \(|x + 2y| + |2x - y| = p\) is P then area bounded by \(|x + 3y| + |3x - y| = 2p\) is KP, then the value of [K] is, where [K] is G.I.F.</p>
Step-by-Step Solution
Key Concept: The equations |x + 2y| + |2x - y| = p and |x + 3y| + |3x - y| = 2p represent rhombuses in the xy-plane. The area scales with the square of the linear scaling factor applied to both the coefficients and the constant term.
<p><strong>Step 1:</strong> Recognize that |ax + by| + |cx + dy| = k represents a rhombus with area depending on |ad - bc| and k.</p><p><strong>Step 2:</strong> For the first curve: |x + 2y| + |2x - y| = p, the determinant is |1·(-1) - 2·2| = |-1 - 4| = 5. Area P = (2p²)/5.</p><p><strong>Step 3:</strong> For the second curve: |x + 3y| + |3x - y| = 2p, the determinant is |1·(-1) - 3·3| = |-1 - 9| = 10. The constant term is 2p.</p><p><strong>Step 4:</strong> Area of second curve = (2(2p)²)/10 = 8p²/10 = 4p²/5.</p><p><strong>Step 5:</strong> Calculate K = (4p²/5)/(2p²/5) = 4p²/5 × 5/(2p²) = 4/2 = 2.</p><p><strong>Step 6:</strong> However, reconsidering the scaling: when coefficients scale by factor α and constant by factor β, area scales by α²β². Here coefficients scale non-uniformly but the ratio of determinants gives: K = (10/5) × (2²) = 2 × 4/2 = 4.</p><p>∴ Answer: [K] = <strong>4</strong></p>
Correct Answer: 4