In the XY-plane, the area, in sq. units, of the region defined by the inequalities y ≥ x+4 and -4 ≤ x²+y²+4(x-y) ≤ 0 is

Question

UPSC Quizroom · Accepted Answer

Inequality: −4 ≤ x² + y² + 4(x − y) ≤ 0

Complete the square: (x + 2)² + (y − 2)² − 8 → 4 ≤ (x + 2)² + (y − 2)² ≤ 8

Circular annulus centered at (−2, 2), inner radius 2, outer radius 2√2

Line y ≥ x + 4 passes through the center → region above line = half annulus

Annulus area = π*(2√2)² − π*2² = 8π − 4π = 4π

Half area = 4π / 2 = 2π

Answer: 2π

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