Question

What should be the distance between an object and a concave mirror of focal length of $20\,cm$ so that a virtual image is produced at the distance of $10\,cm$ from the mirror?
A. $12\,cm$
B. $6.6\,cm$
C. $30\,cm$
D. $25\,cm$

Answer 1

The type of the mirror is necessary to understand the type and the position of image to be formed. Applying values in the equation can give answers, but with appropriate signs. Understanding a mirror is necessary to understand the negativity and positivity of given values. Concave mirror is a convergent mirror as it converges the ray inward upon reflection.

Complete step by step solution:
The signs of points in front of the mirror (on the side of the reflecting surface) are taken as negative. Signs behind the mirror (on the side of the silvered portion) are observed as positive.
For a concave mirror, the focal length of the mirror is in front of the mirror hence the focal length of the concave mirror is considered negative.
$f = ( - 20\,cm)$ = Focal length
While the image formed by the mirror is virtual and is obtained by the mirror behind the reflecting surface hence, the distance of the formed image is considered positive.
$v = 10\,cm$ = Distance of image

For mirror system, relation between focal length, object and image is given by:
$\;\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$
Putting value of focal length and distance from question into equation
$\dfrac{1}{{10}} + \dfrac{1}{u} = \dfrac{1}{{( - 20)}}$
Evaluating equation to obtain value of distance of object from mirror will give,
$u = \left( { - \dfrac{{100}}{{15}}} \right) = \left( { - 6.6667} \right)\,cm$
Negative sign only denotes the direction in which side of mirror the object is present, hence the distance of object from mirror is = $6.67\,cm$

Note:
It is important to properly use the signs of each value. Using opposite signs will change the concept and answer completely. Points in front of the mirror are always negative and ones behind the mirror are taken positive. Concave mirrors are converging mirrors hence most of the images of this mirror are in front of the mirror and are real images. Only two virtual images can be obtained from this mirror, one at infinity and other behind the mirror. For the former condition the object is to be present at the focal point of the mirror and later is obtained when the object is between focal point and mirror.