XÉT VỚI THẤU KÍNH HỘI TỤ:

Hình L1 (ảnh thật)

Ta có:

+ \(\Delta OAB\~\Delta OA'B'\)

\( \Rightarrow \dfrac{{AB}}{{A'B'}} = \dfrac{{OA}}{{OA'}}\) (1)

+ \(\Delta F'IO\~\Delta F'B'A'\)

\( \Rightarrow \dfrac{{I{\rm{O}}}}{{A'B'}} = \dfrac{{F'O}}{{F'A'}}\) (2)

Mà \(IO=AB\)

Từ (1) và (2) ta suy ra:

\(\begin{array}{l}\dfrac{{OA}}{{OA'}} = \dfrac{{F'O}}{{F'A'}} \Leftrightarrow \dfrac{{OA}}{{OA'}} = \dfrac{{F'O}}{{OA' - F'O}}\\ \Leftrightarrow \dfrac{{OA'}}{{OA}} = \dfrac{{OA' - F'O}}{{F'O}} = \dfrac{{OA'}}{{F'O}} - 1\\ \Rightarrow \dfrac{{OA'}}{{OA}} = \dfrac{{OA'}}{{F'O}} - \dfrac{{OA'}}{{OA'}}\\ \Rightarrow \dfrac{1}{{OA}} = \dfrac{1}{{F'O}} - \dfrac{1}{{OA'}}\\ \Rightarrow \dfrac{1}{{F'O}} = \dfrac{1}{{OA}} + \dfrac{1}{{OA'}}\end{array}\)

Hay \(\dfrac{1}{f} = \dfrac{1}{d} + \dfrac{1}{{d'}}\) (ĐPCM)

Hình L2 (ảnh ảo)

+ $\Delta OAB\~\Delta OA'B' \Rightarrow \dfrac{{AB}}{{A'B'}} = \dfrac{{OA}}{{OA'}}\left( 1 \right)$

+ \(\Delta F'OI\~\Delta F'A'B' \Rightarrow \dfrac{{OF'}}{{A'F'}} = \dfrac{{OI}}{{A'B'}}\left( 2 \right)\)

Lại có: \(OI = AB\)

\(\begin{array}{l} \Rightarrow \dfrac{{OA}}{{OA'}} = \dfrac{{{\rm{OF}}'}}{{AF'}} \Rightarrow \dfrac{{OA'}}{{OA}} = \dfrac{{AF'}}{{OF'}} = \dfrac{{OA' + {\rm{OF}}'}}{{{\rm{OF}}'}} = \dfrac{{OA'}}{{{\rm{OF}}'}} + 1\\ \Leftrightarrow \dfrac{{OA'}}{{OA}} = \dfrac{{OA'}}{{OF'}} + \dfrac{{OA'}}{{OA'}}\\ \Rightarrow \dfrac{1}{{OA}} = \dfrac{1}{{OF'}} + \dfrac{1}{{OA'}}\\ \Rightarrow \dfrac{1}{{OF'}} = \dfrac{1}{{OA}} - \dfrac{1}{{OA'}}\end{array}\)

Hay \(\dfrac{1}{f} = \dfrac{1}{d} - \dfrac{1}{{d'}}\)  (ĐPCM)

XÉT VỚI THẤU KÍNH PHÂN KÌ

Ta có:

+ \(\Delta OAB\~\Delta OA'B' \Rightarrow \dfrac{{AB}}{{A'B'}} = \dfrac{{OA}}{{OA'}}\left( 1 \right)\)

+ \(\Delta I{\rm{OF'\~}}\Delta {\rm{B'A'F'}} \Rightarrow \dfrac{{IO}}{{B'A'}} = \dfrac{{{\rm{O}}F'}}{{A'F'}}\left( 2 \right)\)

Mà \(IO = AB\)

Từ (1) và (2) ta suy ra:

\(\begin{array}{l}\dfrac{{OA}}{{OA'}} = \dfrac{{O{\rm{F'}}}}{{A'F'}}\\ \Rightarrow \dfrac{{OA'}}{{OA}} = \dfrac{{A'F'}}{{{\rm{OF}}'}} = \dfrac{{{\rm{OF}}' - OA'}}{{OF'}} = 1 - \dfrac{{OA'}}{{OF'}}\\ \Leftrightarrow \dfrac{{OA'}}{{OA}} = \dfrac{{OA'}}{{OA'}} - \dfrac{{OA'}}{{OF'}}\\ \Rightarrow \dfrac{1}{{OA}} = \dfrac{1}{{OA'}} - \dfrac{1}{{OF'}}\\ \Rightarrow \dfrac{1}{{OF'}} = \dfrac{1}{{OA'}} - \dfrac{1}{{OA}}\end{array}\)

Hay \(\dfrac{1}{f} = \dfrac{1}{{d'}} - \dfrac{1}{d}\) (ĐPCM)