Multiple choice

A flagstaff stands vertically on a pillar,the height of the flagstaff being double the height of the pillar. A man on the ground at a distance finds that both the pillar and the flagstaff subtend equal angles at his eyes. The ratio of the height of the pillar and the distance of the man from pillar, is

  1. $\displaystyle \sqrt{3}:1$
  2. $\displaystyle 1:3$
  3. $\displaystyle 1:\sqrt{3}$
  4. $\displaystyle \sqrt{3}:2$
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C Correct answer
Explanation

Let pillar height be h, flagstaff 2h, distance x. Angle at ground: tan(theta) = h/x. Also tan(2*theta) = (h+2h)/x = 3h/x. Using tan(2*theta) = 2tan(theta) / (1-tan^2(theta)), we get 3h/x = 2(h/x) / (1 - (h/x)^2). Simplifying gives 3 = 2 / (1 - (h/x)^2), so 1 - (h/x)^2 = 2/3, (h/x)^2 = 1/3, h/x = 1/sqrt(3).

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