A converging mirror `M_1` a point source S and a diverging mirror `M_2` are arranged as shown in figure. The source is placed at a distance of 30 cm from `M_1`. The focal length of each of the mirrors is 20 cm. Consider only the images formed by a maximum of two reflections. It is found that one image is formed on the source itself. (a) Find the distance between the two mirrors. (b) Find the location of the image formed by the single reflection from `M_2

Explanation

(a) Image by concave mirror: The direction of incident rays is from right to left
`u=-30cm,f=-20cm`
`(1)/(v)+(1)/(u)=(1)/(f)implies(1)/(v)+(1)/(-30)=(1)/(-20)`
`(1)/(v)=-(1)/(20)+(1)/(30)=(-1)/(60)impliesv=-60cm`
Now whether 60 cm is greater than `(30+d)` or less than `(30+d)`. Since object and image are formed at same place, for this image by convex mirror should be real. The image by convex mirror is real if object is virtual and lie between pole and focus of convex mirror. Hence `60gt(30+d)`. The image of S is formed at `I_1`.
Image by convex mirror: After every reflection, direction of incident ray is reversed, now it is from left to right
`u={60-(30+d)}=30-d,v=-d,f=20`
`(1)/(v)+(1)/(u)=(1)/(f)`
`(1)/(-d)+(1)/(30-d)=(1)/(20)`
`(-30+d+d)/(d(30-d))=(1)/(20)`
`-600+40d=30d-d^2`
`d^2+10d-600=0`
`(d+30)(d-20)=0`
`d=20cm,d=-30` (not possible)
The separation between two mirrors `=30+d=50cm`
(b) `(1)/(v)+(1)/(-20)=(1)/(20)`
`v=10cm`