Question

A tower, x metres, has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant y metres from the foot of the tower. Then find the length of the flagstaff (in metres).

Answer 1

Let the height of the tower AQ be \(x\) metres 

Let the flagstaff PA = h metres 

Let the point R be at a distance y metres from the foot of the tower Q, i.e., RQ = y metres 

Given, ∠PRA = ∠ARQ = α (say) 

Then, in ΔARQ, tan α = \(\frac{x}{y}\)                   ....(i)

In ΔPRQ, tan 2α = \(\frac{h+x}{y}\) ⇒ \(\frac{2\,\text{tan}\,\alpha}{1-\text{tan}^2\,\alpha}\) = \(\frac{h+x}{y}\) 

⇒ \(\frac{\frac{2x}{y}}{\frac{1-x^2}{y^2}}\) = \(\frac{h+x}{y}\)             (From (i))

\(\frac{2xy}{y^2-x^2}\) = \(\frac{h+x}{y}\) ⇒ 2xy² = h(y² – x²) + xy² – x³ ⇒ \(\frac{xy^2+x^3}{y^2-x^2}\) = h 

⇒ h = \(\frac{x(y^2+x^2)}{y^2-x^2}\)m

∴ AP = Length of flagstaff = \(\frac{x(y^2+x^2)}{y^2-x^2}\)m.