Minimum distance for ant avoiding repellents

CAT 2005 Slot 1 · QA · Medium · Geometry

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Four points A, B, C and D lie on a straight line in the X-Y plane, such that AB=BC=CD=1AB = BC = CD = 1 metre. An ant at A wants to reach a sugar particle at D. But there are insect repellents kept at points B and C. The ant would not go within one metre of any insect repellent. The minimum distance in metres the ant must traverse to reach the sugar particle is

  1. A.

    323\sqrt{2}

  2. B.

    1+π1 + \pi

  3. C.

    4π3\frac{4\pi}{3}

  4. D.

    5

Answer

B

Explanation

To avoid going within 1 metre of B and C, the ant must travel along circular arcs of radius 1 m centered at B and C. From A, the ant follows a quarter arc of radius 1 centered at B to point P (angle 90∘90^\circ): arc length =90∘360∘×2π(1)=π2= \frac{90^\circ}{360^\circ} \times 2\pi(1) = \frac{\pi}{2}. From P to Q, it moves along a straight tangent segment of length 11 m. From Q to D, it follows another quarter arc centered at C: arc length =π2= \frac{\pi}{2}. Total minimum distance =π2+1+π2=1+π= \frac{\pi}{2} + 1 + \frac{\pi}{2} = 1 + \pi.

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