A child is standing in front of a straight plane mirror

 

A child is standing in front of a straight plane mirror :- A child is standing in front of a straight plane mirror. His father is standing behind him, as shown in the fig. The height of the father is double the height of the child. What is the minimum length of the mirror required so that the child can completely see his own image and his father’s image in the mirror? Given that the height of the father is 2H.

(A) 3H/2

(B) None of these

(C) H/2

(D) 5H/6

 

Solution :

Key Points:

1. To see his own full image, the mirror must reflect light rays from the lowest point of his body (his own feet) to his eyes (in that case light rays coming from the father’s feet are also covered).
2. To see his father’s full image, the mirror must reflect light rays from the top of the father’s head to the child’s eyes.

Therefore,  let us draw a suitable ray diagram and find the minimum length of the mirror with help of geometry.

Here we have assumed that the eyes of the boy are near his head.

From the above diagram :

In ΔABC,

     …..(1)

In ΔACD,

     …..(2)

From equations (1) and (2),

     …..(3)

Now in ΔABE,

     …..(4)

Similarly in ΔAFG,

     …..(5)

From equations (4) and (5),

     …..(6)

Adding equations (3) and (6), we get

But (x+y) = Length of the mirror (L)

Hence minimum length of the mirror (L_min) required so that the child can completely see his own image and his father’s image is :

     …..(7)

Where H is the height of the boy.