A thin rod of length `f//3` is placed along the optical axis of a concave mirror of focal length f such that its image whichis real and elongated just touches the rod. Calculate the magnification.

Explanation

Image of rod will just touch the rod only when one end of rod is at `'C'` center of curvature. So we have two options as illustrated in following figures.

If `'l'` be the length of image
In case `I`
`(1)/(V_(B))-(3)/((7f))=-(1)/(f)rArrV_(B)=-(7)/(4)f` & `l'=f|(7)/(4)-2|=f//4`
as `l' lt l` this is definitely not the case.
In case `II`
`(1)/(V_(B))-(3)/(5f)=-(1)/(f)`
`rArr V_(B)=-(5f)/(2)` and `l'=f|(5)/(@)-2|=(f)/(2)`
And the required linear magnification `=(l')/(l)=(3)/(2)`