Q.Starting from the geometry of a ray incident on a concave mirror and reflecting through the focus (paraxial approximation), derive the mirror formula $\dfrac{1}{v}+\dfrac{1}{u}=\dfrac{1}{f}$.

Consider an object on the axis of a concave mirror, beyond the centre of curvature, and a ray leaving the tip of the object parallel to the principal axis; after reflection this ray passes through the focus $F$. A second ray from the same point, aimed at the pole $P$, reflects symmetrically about the axis (angle of incidence = angle of reflection). Where these two reflected rays cross fixes the tip of the image. Two pairs of similar triangles can now be read off this geometry: one pair formed by the object, the image, and the pole $P$ (giving the ordinary magnification relation $m=-v/u$ in terms of the heights and distances), and a second pair formed using the focus $F$ and the small gap between the mirror and the point where the parallel ray strikes it (giving a relation between $u$, $v$ and $f$ once the mirror's aperture is treated as small, i.e. in the paraxial approximation, $\sin\theta\approx\tan\theta\approx\theta$). Equating the two independent expressions obtained for the same ratio of heights, and rearranging algebraically using the New Cartesian sign convention (so that $u$, $f$ and $v$ carry the correct signs for a real object and a real image), the equation reduces, after cancelling the common height factor, exactly to $\frac1v+\frac1u=\frac1f$. The derivation goes through unchanged, sign for sign, for a convex mirror or for a virtual image, which is exactly the point of using the sign convention throughout.

Trace one ray parallel to the axis (reflects through F) and one ray to the pole (reflects symmetrically); form two pairs of similar triangles and eliminate the height ratio between them.

• Forgetting the paraxial (small-angle/near-axis) assumption, without which the simple similar-triangles argument does not hold exactly.
• Dropping the sign convention partway through and mixing signed and unsigned distances.
• Confusing this derivation's geometry with the (different) two-ray geometry used for a convex mirror's virtual image.