Question

A tower is $b$ ft. high having a flag staff at its top. The tower and the flag staff subtend equal angles at a point distant $a$ feet from the foot of the tower. Show that the length of the flagstaff is $\frac{b(a^2+b^2)}{a^2-b^2}$.

Answer 1

In $△PAB$,

$\frac{b}{a}=tan\theta$

In $△PAC$,

$\frac{b+x}{a}=tan2\theta$

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

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

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

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

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