Q.By stating the sign conventions and assumptions made, derive mirror formula 1/v + 1/u = 1/f for convex mirror. OR

Using the sign convention (distances measured from the pole, along incident-light direction positive) and similar triangles from a paraxial ray reflected at a convex mirror, the mirror formula 1/v + 1/u = 1/f follows.

Sign convention: All distances are measured from the pole of the mirror. Distances measured in the direction of incident light are taken positive; those measured opposite to it are negative. Heights measured upward (above principal axis) are positive, downward negative.

Assumptions: The aperture of the mirror is small, and rays are paraxial (make small angles with the principal axis), so that the mirror formula is valid to that approximation.

Derivation: Consider an object AB on the principal axis of a convex mirror, at a distance u from the pole P, forming a virtual, erect image A′B′ at distance v (behind the mirror), with focal length f (also behind the mirror, so v and f are positive by convention for a convex mirror while u is negative).

From the geometry of the ray diagram, triangles ABP and A′B′P are similar (for the ray through the pole), giving

$\dfrac{A'B'}{AB}=\dfrac{PB'}{PB}=\dfrac{v}{u}\quad\text{(numerically)}$ …