Multiple choice

At a distance a from the foot of a tower $AB$ of known height $b$ a flagstaff $BC$ and the tower subtend equal angles, height of the flagstaff is

  1. $\dfrac { a ^ { 2 } + b ^ { 2 } } { a ^ { 2 } - b ^ { 2 } }$
  2. $\dfrac { a ^ { 2 } - b ^ { 2 } } { a ^ { 2 } + b ^ { 2 } }$
  3. $\dfrac { a \left( a ^ { 2 } - b ^ { 2 } \right) } { a ^ { 2 } + b ^ { 2 } }$
  4. $\dfrac { b \left( a ^ { 2 } + b ^ { 2 } \right) } { \left( a ^ { 2 } - b ^ { 2 } \right) }$
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D Correct answer
AI explanation

Let the height of the flagstaff be x, making the total height of the structure b + x, while the observer is at distance a from the base. Using the trigonometric condition for equal angles, tan(theta) = b/a and tan(2*theta) = (b + x)/a. Applying the double angle formula, tan(2*theta) = 2(b/a) / (1 - (b/a)^2), which simplifies to 2ab / (a^2 - b^2). Equating this to (b + x)/a and solving for x gives the height of the flagstaff as b(a^2 + b^2) / (a^2 - b^2).

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