To determine the correct graph representing the variation of object distance (u) and image distance ($v$) for a convex mirror, we need to refer to the mirror equation and understand the nature of images formed by a convex mirror.

Mirror Equation

The mirror equation is given by:

1/f=1/v+1/u

For a convex mirror:

The focal length ($f$) is positive.

The object distance (u) is taken as negative (since the object is placed in front of the mirror).

The image distance ($v$) is positive (since the image formed by a convex mirror is virtual and appears to be behind the mirror).

Image Characteristics of a Convex Mirror

The image formed by a convex mirror is always virtual, erect, and diminished.

The image distance ($v$) is always less than the focal length ($f$) and hence always positive but less than $f$.

As the object distance ($u$) increases (i.e., the object moves further away from the mirror), the image distance ($v$) decreases and approaches the focal length asymptotically.

From the given option

Graph (a): This graph shows $v$ increasing asymptotically towards a maximum value as $u$ increases. This is consistent with the behaviour of a convex mirror where $v$ approaches the focal length but never exceeds it.

Graph (b): This is a linear relationship, which is not consistent with the mirror equation for a convex mirror.

Graph (c): This graph shows v decreasing rapidly initially and then levelling off, which is also not consistent with the behaviour of a convex mirror.

Graph (d): This graph shows $v$ decreasing linearly with u, which is not consistent with the mirror equation for a convex mirror.

Hence, the correct answer is option a

To determine the correct graph representing the variation of object distance (u) and image distance ($v$) for a convex mirror, we need to refer to the mirror equation and understand the nature of images formed by a convex mirror.

Mirror Equation

The mirror equation is given by:

1/f=1/v+1/u

For a convex mirror:

The focal length ($f$) is positive.

The object distance (u) is taken as negative (since the object is placed in front of the mirror).

The image distance ($v$) is positive (since the image formed by a convex mirror is virtual and appears to be behind the mirror).

Image Characteristics of a Convex Mirror

The image formed by a convex mirror is always virtual, erect, and diminished.

The image distance ($v$) is always less than the focal length ($f$) and hence always positive but less than $f$.

As the object distance ($u$) increases (i.e., the object moves further away from the mirror), the image distance ($v$) decreases and approaches the focal length asymptotically.

From the given option

Graph (a): This graph shows $v$ increasing asymptotically towards a maximum value as $u$ increases. This is consistent with the behaviour of a convex mirror where $v$ approaches the focal length but never exceeds it.

Graph (b): This is a linear relationship, which is not consistent with the mirror equation for a convex mirror.

Graph (c): This graph shows v decreasing rapidly initially and then levelling off, which is also not consistent with the behaviour of a convex mirror.

Graph (d): This graph shows $v$ decreasing linearly with u, which is not consistent with the mirror equation for a convex mirror.

Hence, the correct answer is option a