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GATE 2026 ME – Question 8

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

Consider a string P of length $l$ that is laid out as a straight-line segment. Another string K is laid out as a semicircular arc with string P as its diameter, as represented in Figure (i). When both the strings are shortened by a length $x$ they can be re-arranged such that the shortened string K forms a full circle with the shortened string P as its diameter, as represented in Figure (ii). The value of $x/l$ is __________.

(i) A straight segment P of length $l$ with a semicircular arc K above it as its diameter. (ii) After shortening, the shortened string K forms a full circle whose diameter is the shortened string P.
  1. $\pi$
  2. $\frac{\pi - 1}{2\pi}$
  3. $\frac{\pi}{2(\pi - 1)}$
  4. $\frac{\pi}{\pi - 1}$

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Correct answer: (C) $\frac{\pi}{2(\pi - 1)}$

Explanation

The semicircle K has length $\frac{\pi l}{2}$. After shortening, P has length $l - x$ and K has length $\frac{\pi l}{2} - x$. A full circle with diameter $l - x$ has circumference $\pi(l - x)$, so $\frac{\pi l}{2} - x = \pi l - \pi x$. This gives $x(\pi - 1) = \frac{\pi l}{2}$, so $\frac{x}{l} = \frac{\pi}{2(\pi - 1)}$.