Question

An object is placed between two parallel plane mirrors shown. Position of third image formed by mirror $M_2$ is a distance x from object then z is:

• A. 2x + 4y
• B. 4x + 2y
• C. 3x + 3y
• D. 4x + 4y

Answer 1

Answer: (B)

4x + 2y

Answer 2

Let the object be O. Let $M_1$ and $M_2$ be two parallel plane mirrors. The distance of the object O from mirror $M_1$ is $y$, and from mirror $M_2$ is $x$. The total distance between the mirrors is $D = x+y$.

Image Formation by Mirror $M_2$:

1. First Image by $M_2$ ($I_{2,1}$): This is the direct image of the object O in mirror $M_2$. The object is at a distance $x$ from $M_2$. So, the first image $I_{2,1}$ is formed at a distance $x$ behind $M_2$. The distance of $I_{2,1}$ from the object O is $x$.

2. Second Image by $M_2$ ($I_{2,2}$): This image is formed by $M_2$ from the first image of $M_1$. The first image of O in $M_1$ ($I_{1,1}$) is formed at a distance $y$ behind $M_1$. The distance of $I_{1,1}$ from $M_2$ is $D + y = (x+y) + y = x+2y$. So, $M_2$ forms its second image $I_{2,2}$ at a distance $x+2y$ behind $M_2$. The distance of $I_{2,2}$ from the object O is $x + (x+2y) = 2x+2y$.

3. Third Image by $M_2$ ($I_{2,3}$): This image is formed by $M_2$ from the second image of $M_1$. The second image of O in $M_1$ ($I_{1,2}$) is formed by $M_1$ from the first image of $M_2$ ($I_{2,1}$). $I_{2,1}$ is at a distance $x$ behind $M_2$. The distance of $I_{2,1}$ from $M_1$ is $D + x = (x+y) + x = 2x+y$. So, $M_1$ forms its second image $I_{1,2}$ at a distance $2x+y$ behind $M_1$. The distance of $I_{1,2}$ from $M_2$ is $D + (2x+y) = (x+y) + (2x+y) = 3x+2y$. So, $M_2$ forms its third image $I_{2,3}$ at a distance $3x+2y$ behind $M_2$. The distance of $I_{2,3}$ from the object O is $x + (3x+2y) = 4x+2y$.

Therefore, the distance $z$ of the third image formed by mirror $M_2$ from the object is $4x+2y$.