GATE AG 2009 • Q4
MCQ • 1 Mark
Section 1: Engineering Mathematics
Calculus
Subtopic: Limit, continuity and differentiability
Question Statement:
A curve is having the equation, r = a(1 − cos θ). The perimeter of the curve between θ = 0 to 2π is
Options:
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(A) 2a
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(B) 4a
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(C) 6a
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(D) 8a
Step-by-Step Solution & Explanation
This is a cardioid r = a(1−cosθ). Arc length in polar form: ds = √[r² + (dr/dθ)²] dθ, with dr/dθ = a sinθ. r² + (dr/dθ)² = a²(1−cosθ)² + a² sin²θ = a²[1 − 2cosθ + cos²θ + sin²θ] = a²[2 − 2cosθ] = 2a²(1−cosθ). Using the half-angle identity 1−cosθ = 2sin²(θ/2): r²+(dr/dθ)² = 4a² sin²(θ/2), so ds = 2a|sin(θ/2)| dθ. Perimeter = ∫[0 to 2π] 2a sin(θ/2) dθ = 2a·[-2cos(θ/2)] from 0 to 2π = 2a·(2+2) = 8a. This matches option (D). Numerical integration confirms 8a exactly.
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