A short linear object of length b lies along the axis of a concave mirror or focal length f at a distance u from the pole of the mirror. The size of the image is approximately equal to

Options

A. `(lf)/((d+d))`
B. `(d+f)/(fl)`
C. `(lf^(2))/((d+f)^(2))`
D. `(l(d+f)^(2))/(f^(2))`

Answer

C

Explanation

The object is short and it lies on the axis of the mirror. Hence, differentiate for small variations of v and u.
For a mirror. `(1)/(v) + (1)/(u) = (1)/(f)`
`rArr-(dv)/(v^(2))-(du)/(u^(2))=0`
`rArr(du)/(v^(2))=-(du)/(u^(2))`
`(dv)/(du)=-(v^(2))/(u^(2)) " " (i)`
Again `(1)/(v)+(1)/(u)=(1)/(f)`
`rArr (u)/(v)=(u)/(f)-1=(u-f)/(f)`
`rArr(u)/(v)+1=(u)/(f)`
`rArr(v)/(u)=(f)/((u-f))`
`rArr (v)/(u) = - (f)/((d+f)) " " (ii)`
From Eqs. (ii) and (i)
`(dv)/(du)=-((f)/(d+f))^(2) " " ("Given du"=f)`
`(dv)/(l)=-((f)/(d+f))^(2)`
`rArr dv=(lf^(2))/((d+f)^(2))` [numerically]