Mathematics

Mathematics

Problem

An arc PQ of a circle of radius R subtends an angle θ\theta at the centre of circle. If A(θ)A(\theta) is the area bounded by the chord and the arc and B(θ)B(\theta) is the area bounded by tangent at P & Q with the arc. Find lim⁡θ→0A(θ)B(θ)\lim_{\theta \rightarrow 0} \frac{A(\theta)}{B(\theta)} .
Intuitive explanation

The key is recognizing that both A(θ)A(\theta) and B(θ)B(\theta) are areas involving the arc PQ, but measured against different boundaries—a chord versus two tangent lines. As θ→0\theta \to 0, both areas shrink, so you need to find their limiting ratio using Taylor expansions. The sector area 12R2θ\frac{1}{2}R^2\theta will appear in both expressions; the trick is identifying what each region actually encloses and expanding carefully to see which terms survive in the limit.

Intuitive explanation

The key is recognizing that both A(θ)A(\theta) and B(θ)B(\theta) are areas involving the arc PQ, but measured against different boundaries—a chord versus two tangent lines. As θ→0\theta \to 0, both areas shrink, so you need to find their limiting ratio using Taylor expansions. The sector area 12R2θ\frac{1}{2}R^2\theta will appear in both expressions; the trick is identifying what each region actually encloses and expanding carefully to see which terms survive in the limit.

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