(a) A point source S is placed at a distance of 15 cm from a converging lens of focal length 10 cm on its principal axis, where should a diverging mirror of focal length 12 cm be placed so that a real image is formed on the source itself?
(b) A diverging lens of focal length 20 cm and a converging mirror of focal length 10 cm are placed coaxially at separation of 5 cm. Where should an object be placed so that a real image is formed at the object itself?
(c) A converging lens of focal length 12 cm and a diverging mirror of focal length 7.5 cm are placed 5.0 cm apart with the principal axis coinciding where should an object be placed so that its image falls on itself?

Explanation

(a) If final image and object are at same place by (lens`+` mirror) combination for this rays after refraction must be incident normally on the mirror i.e., rays must be passed through centre of curvature of mirror. In brief, image by lens shoudl be formed on centre of curvature of mirror.
Image by lens should be at centre of curvature of mirror.
`u=-15cm,f=10cm,v=(d+24)cm`
`(1)/(v)-(1)/(u)=(1)/(f)implies(1)/(v)=(1)/(u)+(1)/(f)=(1)/(-15)+(1)/(10)=(1)/(30)`
`v=30=d+24impliesd=6cm`
(b) Image by lens should be formed at centre of curvature of mirror which is 15 cm left to lens.
`u=-x,v=-15,f=-20cm`
`(1)/(v)-(1)/(u)=(1)/(f)`
`(1)/(-15)-(1)/(-x)=(1)/(-20)implies(1)/(x)=(1)/(15)-(1)/(20)=(1)/(60)`
`x=60cm`
(c) `u=-x,v=20cm,f=12cm`
`(1)/(20)-(1)/(-x)=(1)/(12)implies(1)/(x)=(1)/(12)-(1)/(20)=(5-3)/(60)`
`x=30cm`