Question

Let $10$ vertical poles standing at equal distances on a straight line, subtend the same angle of elevation $\alpha$ at a point $O$ on this line and all the poles are on the same side of $O$. If the height of the longest pole is $h$ and the distance of the foot of the smallest pole from $O$ is $a$; then the distance between two consecutive poles, is

• A. $\frac{h\sin \alpha +a\cos \alpha }{9\cos \alpha }$
• B. $\frac{h\cos \alpha -a\sin \alpha }{9\sin \alpha }$
• C. $\frac{h\sin \alpha +a\cos \alpha }{9\sin \alpha }$
• D. $\frac{h\cos \alpha -a\sin \alpha }{9\cos \alpha }$

Answer 1

The correct option is B $\frac{h\cos \alpha -a\sin \alpha }{9\sin \alpha }$

Since all $10$ poles are subtending equal angles at $O$. Let the distance between two consecutive poles $=d$.
Distance form $O$ to the smallest pole $=a$
Total base distance in right angled triangle $=a+9d$
$\tan \alpha =\frac{h}{a+9d}$
$9d\tan \alpha +a\tan \alpha =h$
$d=\frac{h-a\tan \alpha }{9\tan \alpha }$
$=\frac{h-a\frac{\sin \alpha }{\cos \alpha }}{9\frac{\sin \alpha }{\cos \alpha }}$

$=\frac{h\cos \alpha -a\sin \alpha }{9\sin \alpha }$