Question

A point object O is placed midway between the concave mirrors, distance d apart. What is the value of d for which object and images coincide? Each mirror has focal length F.

• A. F, 2 F
• B. 2 F, 3 F
• C. F, 4 F
• D. 2 F, 4 F

Answer 1

The correct option is D 2 F, 4 F

The distances will be denoted in boldface and the values to be used in the formula (by sign convention) will be in italics.
For the first reflection from mirror in the right,

$u=\frac{d}{2},u=-u,f=-F$

$\frac{1}{v}=\frac{1}{f}-\frac{1}{u}=\frac{-1}{F}-\frac{-2}{d}$

$v=\frac{dF}{2F-d}$

$v=\frac{dF}{d-2F}$

For the mirror opposite to the first one, $u_1=d-v$

$u_1=\frac{d^2-3dF}{d-2F}$

We already know that $v_1=\frac{d}{2}$

$u_1=-u_1,v-v_1,f=-F$

So we have

$\frac{-2}{d}+\frac{d-2F}{3dF-d^2}=\frac{-1}{F}$

Solving this we get $d-2F=0$ or $d^2-4dF=0$

So $d=2F$ or $d=4F$ or $d=0$

But $d$ cannot be zero (from the figure)

So $d=2F$ or $d=4F$
Note that $v$ or $u$ can be $\pm \infty$