At a distance 'a' from the foot of a tower of height 'b' ,the tower and the flagstaff at the - Maths - Some Applications of Trigonometry

Question

At a distance 'a' from the foot of a tower of height 'b' ,the
tower and the flagstaff at the top of the tower make equal angles
Show that the height of the flagstaff is b[ a2+
b2] / a2 - b2

ramandeep.singh · Accepted Answer

h = height of flagstaff b = height of tower a = distance between tower and point of observation

Here, we make use of the formula $\tan (2 α) = \frac{2 \tan α}{1 - \tan^{2} α}$ In right Î”DBC $\tan θ = \frac{b}{a} (1)$ In right Î”ABC $\tan (θ + θ) = \frac{h + b}{a} \Rightarrow \tan (2 θ) = \frac{h + b}{a} \Rightarrow \frac{2 \tan θ}{1 - \tan^{2} θ} = \frac{h + b}{a}$ Using equation (1), we get $\frac{2 (\frac{b}{a})}{1 - (\frac{b}{a})^{2}} = \frac{h + b}{a} \Rightarrow \frac{2 a b}{a^{2} - b^{2}} = \frac{h + b}{a} \Rightarrow 2 a^{2} b = a^{2} h + a^{2} b - b^{2} h - b^{3} \Rightarrow h (a^{2} - b^{2}) = b (a^{2} + b^{2}) ∴ h = \frac{b (a^{2} + b^{2})}{(a^{2} - b^{2})}$ Hence, proved

At a distance 'a' from the foot of a tower of height 'b' ,the tower and the flagstaff at the top of the tower make equal angles. Show that the height of the flagstaff is b[ a2+ b2] / a2 - b2 .

At a distance 'a' from the foot of a tower of height 'b' ,the tower and the flagstaff at the top of the tower make equal angles. Show that the height of the flagstaff is b[ a²+ b²] / a² - b^{2 .}