question_answer

A ladder rests against a wall at an angle a to the horizontal. Its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle \[\text{ }\!\!\beta\!\!\text{ }\] with the horizontal. Then a is equal to

A)  \[b\tan \frac{1}{2}(\alpha +\beta )\]     

B)  \[b\tan \frac{1}{2}(\alpha -\beta )\]

C)  \[a\tan \frac{1}{2}(\alpha -\beta )\]      

D)  None of these

Correct Answer: A

Solution :

Let Z be the length of the ladder Now, \[a+OA-OB=l\cos \beta -l\cos \alpha \] and   \[b=OP-OQ=l\sin \alpha -l\sin \beta \] \[\therefore \]                  \[\frac{a}{b}=\frac{\cos \beta -\cos ga}{\sin \alpha -\sin \beta }\] \[\Rightarrow \]\[\frac{a}{b}=\frac{2\sin \left( \frac{\alpha +\beta }{2} \right)\sin \left( \frac{\alpha -\beta }{2} \right)}{2\cos \left( \frac{\alpha +\beta }{2} \right)\sin \left( \frac{\alpha -\beta }{2} \right)}\] \[\Rightarrow \]\[a=b\tan \left( \frac{\alpha -\beta }{2} \right)\]