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

As in question, image touches the rod, i.e., image and object coincides, hence one end of the rod should be at the center of curvature. It is also written that image is enlarged, it indicates that the orientation of rod should be toward focus then only we can get enlarged image along the principal axis. Let l be the length of the image.
Then, `m=l/(f//3) rArr l=(mf)/3`
Also, one end of the image coincides with the object, `u^(')=2f`.
Now, `u^(')= u+f/3 rArr u=2f-f/3=(5f)/3`
`v=-(u+f/3+(mf)/3).`
Putting in mirror formula, we get
`1/(u+f//3+mf//3)+1/u=1/f rArr 3/(5f+f+mf)+3/(5f)=1/f`
`rArr (1)/(M+6)=(2)/(15) rArr m=(3)/(2)`