Question

Assertion ($A$): ladder rests against a wall at an angle ${30}^0$ to the horizontal. Its foot is pulled away through a distance $x$' so that it slides a distance $y$' down the wall finally making an angle ${60}^0$ with the horizontal then $x=y$.
Reason ($R$): $A$ ladder rests against a wall at angle $\alpha$ to the horizontal. Its foot is pulled a way through a distence $a$' so that it slides a distence $b$' down the wall, finally making an angle $\beta$ with the horizonal then $\tan \left(\frac{\alpha +\beta }{2}\right)=b/a$

• A. Both $A$ and $R$ are ture and ` $R$' is the correct
  explanation of $A$
• B. Both $A$ and $R$ are true and ` $R$' is not correct
  explanation of $A$
• C. $A$ is true but ` $R$' is false
• D. $A$' is false but $R$' is true.

Answer 1

The correct option is C $A$ is true but ` $R$' is false

$AB=A^{\prime }{B}^{\prime }=l$

$\sin \alpha =\frac{AC}{l}$

$\sin \beta =\frac{A^{\prime }C}{l}$

$\sin \alpha -\sin \beta =\frac{b}{l}$

$\cos \beta -\cos \alpha =\frac{a}{l}$

$\frac{2\sin \left(\frac{\alpha -\beta }{2}\right)\cos \left(\frac{\alpha +\beta }{2}\right)}{2\sin \left(\frac{\alpha +\beta }{2}\sin \left(\frac{\alpha -\beta }{2}\right)\right)}=\frac{b}{a}$

$\cot \left(\frac{\alpha +\beta }{2}\right)=\frac{b}{a}$