Question

A tower of height b and a flagstaff on top of tower subtend equal at a point on the ground, distant a from the base of the tower. The height of the flagstaff is

• A. $\frac{b^2}{a+b}$
• B. $\frac{b(a+b)}{a-b}$
• C. $\frac{b^2}{a-b}$
• D. $b\left(\frac{a^2+b^2}{a^2-b^2}\right)$

Answer 1

The correct option is D $b\left(\frac{a^2+b^2}{a^2-b^2}\right)$

Let the height of the flagstaff be 'x'

In $\Delta ABD$

$tan\theta =\frac{b}{a}$ ……$\left(1\right)$

In $\Delta ACD\Rightarrow \tan 2\theta =\frac{b+x}{a}$

$\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }=\frac{b+x}{a}$

Substituting the value of $\tan \theta$ as $\frac{b}{a}$ (equation $\left(1\right)$)

$\Rightarrow \frac{\frac{2b}{a}}{1-\frac{b^2}{a^2}}=\frac{b+x}{a}$

$\Rightarrow \frac{2b.a^2}{a(a^2-b^2)}=\frac{b+x}{a}\Rightarrow x=\frac{2a^{2}b}{a^2-b^2}-b$

$\Rightarrow x=\frac{2a^{2}b-a^{2}b+b^3}{(a^2-b^2)}=\frac{a^{2}b+b^3}{a^2-b^2}=b\left(\frac{a^2+b^2}{a^2-b^2}\right)$

[D].