Four identical mirror are made to stand vertically to form a square arrangement as shown in a top view. A ray starts from the midpoint M of mirror AD and after two reflections reaches corner D. Then, angle `theta` must be

Options

A. `tan^(-1)(0.75)`
B. `cot^(-1)(0.75)`
C. `sin^(-1)(0.75)`
D. `cos^(-1)(0.75)`

Answer

B

Explanation

The ray starting from point M at an angle `theta` reaches the corner D at the right along a parallel path. Let a be the length of the side.
From figure,
`tan theta = (x)/((a//2)) "…."(i)`
`tan theta = (a-x)/(y) "….."(ii) , tan theta = (a)/(a-y) "...."(iii)`
From (i) and (ii) we get
`(2x)/(a) = (a-x)/(y)` or `2xy = a^(2) - xa"......"(iv)`
From (ii) and (iii), we get
`(a-x)/(y) = (a)/(a-y) rArr 3xy = 2ay` (Using (iv))
`x = (2a)/(3)`
Substituting this value of x in equation (i), we get
`tan theta = ((2a//3))/((a//2)) = (4)/(3), :. cot theta = (1)/(tan theta) = 3/4`
or `theta = cot^(-1)(0.75)`