When an object is placed at the center of curvature (C) of a concave mirror, what is the nature of the image formed?

The correct option is Real and inverted.

Step-by-Step Explanation:

1. Understanding Concave Mirrors:
A concave mirror is a spherical mirror whose reflecting surface is curved inwards (towards the center of the sphere). When parallel light rays fall on a concave mirror, they converge at a single point called the principal focus (F).

2. Mirror Formula and Ray Tracing Analysis:
Let us use the mirror formula to analyze the position and magnification of the image formed when an object is placed at the center of curvature (C):

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

Where:
• $f$ is the focal length of the mirror.
• $u$ is the object distance.
• $v$ is the image distance.

According to the Cartesian sign convention for a concave mirror:
• Focal length, $f=-f$
• Object distance at the center of curvature ($C=2f$), $u=-2f$

Substituting these values into the mirror formula:

$-\frac{1}{f}=\frac{1}{v}-\frac{1}{2f}$

$\frac{1}{v}=-\frac{1}{f}+\frac{1}{2f}=-\frac{1}{2f}$

$v=-2f$

This shows that the image is formed at the center of curvature (C) in front of the mirror.

3. Determining Image Characteristics (Magnification & Nature):
Magnification ($m$) is given by the ratio:

$m=-\frac{v}{u}$

Substituting $v=-2f$ and $u=-2f$:

$m=-\frac{-2f}{-2f}=-1$

• The negative sign of magnification indicates that the image is real and inverted (formed in front of the mirror, below the principal axis).
• The absolute value of magnification ($\left|m\right|=1$) indicates that the image is of the same size as the object.

Thus, the image formed is Real and inverted.