$(a)$ State the relation between object distance,image distance and the focal length of a spherical mirror.
$(b)$ Draw a ray diagram to show the image formed by a concave mirror when an object is placed between the pole and focus of the mirror.
$(c)$ $A$ concave mirror of focal length $15\, cm$ forms an image of an object kept at a distance of $10\, cm$ from the mirror. Find the position,nature and size of the image formed by it.

(N/A) The mirror formula is given by $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$,where $f$ is the focal length,$v$ is the image distance,and $u$ is the object distance.
$(b)$ The ray diagram shows the object placed between the pole $(P)$ and the focus $(F)$ of a concave mirror,resulting in a virtual,erect,and magnified image behind the mirror.
$(c)$ Given:
Focal length of concave mirror,$f = -15\, cm$
Object distance,$u = -10\, cm$
Using the mirror formula: $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$
$\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{-15} - \frac{1}{-10} = -\frac{1}{15} + \frac{1}{10} = \frac{-2 + 3}{30} = \frac{1}{30}$
Therefore,$v = +30\, cm$.
Since $v$ is positive,the image is formed $30\, cm$ behind the mirror.
The image is virtual,erect,and magnified (as magnification $m = -\frac{v}{u} = -\frac{30}{-10} = +3$).