2023 · Q33
AB is a straight road leading to the foot P of a tower of height h. Q is at distance x from P and R is at a distance y from Q (R is farther from P than Q; R, Q are on the same side). The angle of elevation of the top of the tower at Q is twice of that at R. Which one of the following is correct?
Elementary MathematicsTrigonometry
- Ax = y
- Bx < y
- Cx > y
- DCannot be concluded due to insufficient data
Explanation
Let the angle at R be φ and at Q be 2φ, with tanφ = h/(x+y) and tan2φ = h/x. Using tan2φ = 2tanφ/(1 − tan²φ) with t = h/(x+y): h/x = 2t/(1−t²) = 2h(x+y)/((x+y)² − h²). Cancelling h and cross-multiplying gives (x+y)² − h² = 2x(x+y), which simplifies to y² = x² + h². Since h > 0, y² > x² and hence y > x, i.e. x < y. The answer is (B).