Question

A concave mirror of radius of curvature R produces a real image n times the size of the object then the distance of the object from the mirror is

Answer 1

Given that:
f = -R/2
m=-n (beacause a real image of a real object is always inverted)
magnification = f/(f-u) = -(R/2)/(-R/2-u) = -n
Implies that R/(R+2u)= -n
Implies that R = -Rn - 2nu
Implies that R+Rn = -2nu
Implies that u = (-R - Rn)/2n
Implies that u = -R(n+1)/2n

Answer 2

Given:

The radius of curvature of the concave mirror = R

The ratio of the size of the real image to that of the object = n

To Find:

The distance of the object from the mirror (u)

Solution:

The distance of the object from the mirror is (4) - (n + 1)R / 2

The magnification formula for a concave mirror = m = hi / ho = -v / u

Here hi and ho are the heights of the image and the object respectively.

According to the question,

hi / ho = -n

or -v / u = -n     (-n because the image is real)

The mirror formula is: 1/f = 1/v + 1/u

and f = R/2

Substituting,

2/R = 1/v + 1/u

or 1/v = 2/R - 1/u

Taking LCM on the Right Hand Side,

1/v = 2u - R / Ru

or v = Ru / 2u - R

Substituting the value of v in the magnification formula,

$\frac{-\frac{Ru}{2u-R} }{u} = -n$

or -Ru = (2u-R) (-un)

or -R = -2un + Rn

or -2un = -Rn - R

or u = - (n + 1) R / 2n