The convex surface of a thin concavo-convex lens of glass of R.l. 1.5 has a radius of curvature 20 cm. The concave surface has a radius of curvature 60 cm. The convex side is silvered and placed horizontally.
Q Where should a pin be placed on the axis such that its image is formed on the same point.

Explanation

The effective focal length F of the combination is given as `- 1/F = 1/f_(g) - 1/f_(m) + 1/f_(g)` because refraction takes place twice with the curved surface and one reflection takes places with the mirror.
`implies - 1/F = 2/f_(g) - 1/f_(m)`
Since `1/f_(g) = (n-1)(1/R_(1) - 1/R_(2))` & `f_(m)= R_(1)/2` , where `R_(1)` and `R_(2)` are the radii of curvature of lower and upper surface the lens. Putting `R_(1)=20cm, R_(2)=60cm, n_(2)=1.5`
`f_(m)= (-R_(1))/2 = (-20)/2 implies f_(m) = -10`
we obtain `f_(g)`= +60cm
`implies - 1/F = 2/60 - 1/(-10)`
`implies F=-7.5 cm` (concave mirror)
For the image to be formed at the same points, the object distance
`l = 2F=-15cm`
Hence, the object should be placed at a distance of 15 cm from the lens on optic axis.