A concave lens of focal length 20 cm is placed 15 cm in front of a concave mirror of radius of curvature 26 cm and further 10 cm away from the lens is placed an object. The principal axis of the lens and the mirror are coincident and the object is on the axis Find the position and nature of the image.

Explanation

Image by lens: `u=-10cm,f=-20cm,v=?`
`(1)/(v)-(1)/(u)=(1)/(f)implies(1)/(v)-(1)/(-10)=(1)/(-20)`
`(1)/(v)=(1)/(-10)-(1)/(20)`
`v=(-20)/(3)cm`
Image of `O` is formed at `I_1`
Image by mirror:
`u'=-((20)/(3)+15)=-(65)/(3)cm,f'=-13cm,v'=?`
`(1)/(v')+(1)/(u')=(1)/(f')rArr(1)/(v')+(1)/(-65//3)=(1)/(-13)`
`(1)/(v')=-(1)/(13)+(3)/(65)=(-5+3)/(65)`
`v'=(-65)/(2)cm`
Image of `I_(1)` is formed at `I_(2)`
Again by lens: Now direction of incident rays reversed
`u_(0)=+((65)/(2)-15)=(35)/(2)cm,f=-20cm,v_(0)=?`
`(1)/(v_0)-(1)/(u_0)=(1)/(f)implies(1)/(v_0)-(1)/((+35)/(2))=(1)/(-20)`
`(1)/(v_0)=-(1)/(20)+(2)/(35)=(-35+40)/(700)`
`v_(0)=140cm`
"Image of" `I_(2)` "is formed at" `I_(2)` Final image is formed at distance 140 cm from lens on the same side of object.