Derive Concave Mirror formula in case of Spherical refracting surfaces 

Solution:
Mirror Formula

The above figure shows the ray diagram for image formation by a concave mirror.

In figure, triangles and ENF are similar.

As the aperture of the concave mirror is small, the points N and P lie very close to each other.

∴NF ≈ PF and NE = AB

Since all the distances are measured from the pole of the concave mirror, we have

Also, triangles ABP and are similar.

From equations (i) and (ii), we obtain

Applying the new Cartesian sign conventions, we have

PA = − u (distance of object is measured against incident ray)

= − v (distance of image is measured against incident ray)

PF = − f (focal length of concave mirror is measured against incident ray)

Substituting these values in equation (iii),

We have

or