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Circular Wall and Gate Sightline

CAT 2001 Slot 1 · Quantitative Ability · Hard · Geometry

This is a hard Quantitative Ability question from the CAT 2001 Slot 1 paper. It tests Geometry. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and west. A house stands outside the city, 3 km3\text{ km} north of the north gate, and it can just be seen from a point 9 km9\text{ km} east of the south gate. What is the diameter of the wall that surrounds the city?

  1. A.

    6 km

  2. B.

    9 km

  3. C.

    12 km

  4. D.

    None of these

Answer

B

Explanation

Let center of circle be origin (0,0)(0,0) and radius be rr. North gate N=(0,r)N = (0, r), South gate S=(0,−r)S = (0, -r). House position A=(0,r+3)A = (0, r+3). Point of view P=(9,−r)P = (9, -r). Line connecting A(0,r+3)A(0, r+3) and P(9,−r)P(9, -r) is tangent to the circle x2+y2=r2x^2+y^2=r^2. Slope of line AP=−r−(r+3)9−0=−2r−39AP = \frac{-r - (r+3)}{9 - 0} = \frac{-2r-3}{9}. Using distance from origin (0,0)(0,0) to line AP=rAP = r gives r=4.5 kmr = 4.5\text{ km}. Diameter =2r=9 km= 2r = 9\text{ km}.

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