# A concave mirror of focal length f produces an image n times the size of the object. If the image is real then the distance of the object is:

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A concave mirror of focal length f produces an image n times the size of the object. If the image is real then the distance of the object is:
1. (n – 1)/ f
2. (n + 1)/f
3. (n+1)f/n
4. None of these

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Correct Answer - Option 3 : (n+1)f/n

CONCEPT:

• Concave mirror: If the inner surface of the spherical mirror is the reflecting surface, then it is called a concave mirror. It is also called a focusing mirror/converging mirror.
• The size of the image produced by these mirrors can be larger or smaller than the object, depending upon the distance of the object from the mirror.
• The concave mirror can form both real as well as virtual images of any object.
• Mirror formula: The expression which shows the relation between object distance (u)image distance (v), and focal length (f) is called mirror formula.

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Linear magnification (m):

• It is defined as the ratio of the height of the image (hi) to the height of the object (ho).

$m = \frac{{{h_i}}}{{{h_o}}}$

• The ratio of image distance to the object distance is called linear magnification.

$m = \frac{{image\;distance\;\left( v \right)}}{{object\;distance\;\left( u \right)}} = - \frac{v}{u}$

• positive value of magnification means virtual an erect image.
• negative value of magnification means a real and inverted image.

CALCULATION:

Given f = focal length and  m = -n (negative sign shows that the image will be inverted to the object)

• We know that in the concave mirror the real image will always be inverted to the object.

Since,

$⇒m= {{h'} \over h}=\frac{-v}{u}=-n$

⇒ v = nu     -----(1)

The mirror formula is given as,

$\Rightarrow \frac{1}{f}=\frac{1}{v}+\frac{1}{u}$     -----(2)

By equation 1 and equation 2,

$\Rightarrow \frac{1}{f}=\frac{1}{nu}+\frac{1}{u}$

$\Rightarrow \frac{1}{f}=\frac{n+1}{nu}$

$\Rightarrow u=\frac{(n+1)}{n}f$

• Hence, option 3 is correct.