A man is standing exactly midway between a wall and a mirror

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A man is standing exactly midway between a wall and a mirror

Question :- A man is standing exactly midway between a wall and a mirror and he wants to see the full height of the wall (behind him), in a plane mirror (in front of him). If the height of the wall is H, then the minimum length of the mirror should be

(A) H/4

(B) 2H/3

(C) H/3

(D) H/5

Solution :

Key Point: To see the full image of wall, the mirror must reflect light rays from the lowest point and the highest point of wall to his eyes.

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

In ΔABC,

$\displaystyle tan\theta =\frac{y}{2d}$ …..(1)

In ΔEDC,

$\displaystyle tan\theta =\frac{q}{d}$ …..(2)

From equations (1) and (2),

$\displaystyle \frac{y}{2d}=\frac{q}{d}$

$\displaystyle y=2q$ …..(3)

Now in ΔFGH,

$\displaystyle tan\alpha =\frac{x}{2d}$ …..(4)

Similarly in ΔEFD,

$\displaystyle tan\alpha =\frac{p}{d}$ …..(5)

From equations (4) and (5),

$\displaystyle \frac{x}{2d}=\frac{p}{d}$

$\displaystyle x=2p$ …..(6)

Length of the wall,

L_w= x + L_m + y

$\displaystyle {{L}_{w}}=2p+(p+q)+2q=3p+3q=3(p+q)$

$\displaystyle \Rightarrow {{L}_{w}}=3{{L}_{m}}$

Therefore, the minimum length of the mirror should be

$\displaystyle {{\left( {{L}_{mirror}} \right)}_{min}}=\frac{{{L}_{w}}}{3}$

Hence option (3) is the correct option.

Class 12th NCERT

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