# Derive mirror formula of convex mirrorâ€‹

Mirror formula

### The following sign convention is used for measuring various distances in the ray diagrams of spherical mirrors:

• All distances are measured from the pole of the mirror.
• Distances measured in the direction of the incident ray are positive and the distances measured in the direction opposite to that of the incident rays are negative.
• Distances measured above the principal axis are positive and that measured below the principal axis are negative.

\begin{aligned} &\text { In } \Delta A B C \text { and } \Delta A^{\prime} B^{\prime} C\\ &\Delta A B C \sim \Delta A^{\prime} B^{\prime} C[ \text { AA similarity }]\\ &\frac{A B}{A^{\prime} B^{\prime}}=\frac{A C}{A^{\prime} C^{\prime}} \ldots(1) \end{aligned}

Similarly, In $\triangle{FPE}$ and $\triangle A^{\prime} B^{\prime} F$

$\begin{array}{l}{\frac{E P}{A^{\prime} B^{\prime}}=\frac{P F}{A^{\prime} F}} \\ \\ {\frac{A B}{A^{\prime} B^{\prime}}=\frac{P F}{A^{\prime} F}[\mathrm{AB}=\mathrm{EP}] \cdots(\mathrm{II})}\end{array}$

From (i) &(ii)

$\begin{array}{l}{\frac{A C}{A^{\prime} C}=\frac{P F}{A^{\prime} F}} \\ \\ {=>\frac{A^{\prime} C}{A C}=\frac{A^{\prime} F}{P F}} \\ \\ {=>\frac{\left(C P-A^{\prime} P\right)}{(A P-C P)}=\frac{\left(A^{\prime} P-P F\right)}{PF} }\end{array}$

Now, $PF = -f$  ;  $CP=2PF=-2f ; AP= -uAP=-u \ \text{and} \ A'P=-v$

Put these value in above relation:

$\begin{array}{l}{\Longrightarrow \frac{[(-2 f)-(-v)]}{(-u)-(-2 f)}=\frac{[(-v)-(-f)]}{(-f)}} \\ \\ {\Longrightarrow u v=f v+u f} \\ \\ {\Longrightarrow \frac{1}{f}=\frac{1}{u}+\frac{1}{v}}\end{array}$

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