You are asked to design a sharing mirror assuming that a person keeps it $ 10\text{ }cm $ from his face and views a magnified image of the face at the closest comfortable distance of $ 25\text{ }cm $ . The radius of curvature of the mirror would be:
(A) $ -60\text{ }cm $
(B) $ 30\text{ }cm $
(C) $ -24\text{ }cm $
(D) $ 24\text{ }cm $

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Hint: As the image formed is magnified and erect, this means the image is virtual and so the value of v is taken to be positive. Substituting the values in the mirror formula, we can find the value of focal length and hence, radius of curvature.

Complete step by step solution
We know that:
 $ $ $ u=-10\text{ }cm $
Now, the position of the object is $ 10\text{ }cm $ left of mirror and position of image is $ 25-10=15\text{ }cm $ right of mirror.
 $ \therefore v=15\text{ }cm $
We know that the mirror formula is given by:
 $ \dfrac{1}{v}+\dfrac{1}{u}=\dfrac{1}{f} $
 $ \begin{align}
  & \dfrac{1}{15}+\dfrac{1}{\left( -10 \right)}=\dfrac{1}{f} \\
 &\Rightarrow \dfrac{1}{f}=\dfrac{-10+15}{\left( -150 \right)} \\
 &\Rightarrow \dfrac{1}{f}=\dfrac{+5}{\left( -150 \right)} \\
 &\Rightarrow \dfrac{1}{f}=\dfrac{-1}{30} \\
\end{align} $
Or $ f=-30\text{ }cm $
Radius of curvature R $ =2f $
$ =2\times \left( -30 \right)=-60\text{ }cm $

A concave mirror is used in the case of shaving mirror because when the concave mirror is placed very close to the object, a magnified and virtual image is obtained.
Concave mirror is also known as a converging mirror. It has a reflecting surface that is recessed inward (away from the incident light). These mirrors reflect light inward to one focal point.
There is several sign convention for concave mirror which are given as follows:
When an image is formed in front of the mirror, the distance of the image is taken as negative. When an image is formed behind the mirror, the distance of the image is taken as positive.
Since the center of curvature and focus lie in front of the concave mirror, signs of radius of curvature and focal length are taken as negative in case of concave mirror.
Height of image is taken as positive in case of erect image and taken as negative in case of inverted image.