Đáp án:
a) \(\dfrac{1}{f} = \dfrac{1}{{d'}} + \dfrac{1}{d}\)
b) \(\dfrac{1}{f} = - \dfrac{1}{{d'}} + \dfrac{1}{d}\)
c) \( - \dfrac{1}{f} = - \dfrac{1}{{d'}} + \dfrac{1}{d}\)
Giải thích các bước giải:
a) Ta có:
\(\begin{array}{l}
\dfrac{{AB}}{{A'B'}} = \dfrac{{OA}}{{OA'}} \Rightarrow \dfrac{h}{{h'}} = \dfrac{d}{{d'}}\\
\dfrac{{AB}}{{A'B'}} = \dfrac{{OI}}{{A'B'}} = \dfrac{{OF'}}{{OA' - OF'}} = \dfrac{f}{{d' - f}}\\
\Rightarrow \dfrac{h}{{h'}} = \dfrac{d}{{d'}} = \dfrac{f}{{d' - f}}\\
\Rightarrow d.d' - df = d'f\\
\Rightarrow \dfrac{1}{f} = \dfrac{1}{{d'}} + \dfrac{1}{d}\\
\Rightarrow d' = \dfrac{{df}}{{d - f}}\\
\Rightarrow \dfrac{h}{{h'}} = \dfrac{d}{{d'}} = \dfrac{d}{{\dfrac{{df}}{{d - f}}}} = \dfrac{{d - f}}{f}
\end{array}\)
b) Ta có:
\(\begin{array}{l}
\dfrac{{AB}}{{A'B'}} = \dfrac{{OA}}{{OA'}} \Rightarrow \dfrac{h}{{h'}} = \dfrac{d}{{d'}}\\
\dfrac{{AB}}{{A'B'}} = \dfrac{{OI}}{{A'B'}} = \dfrac{{OF'}}{{OA' + OF'}} = \dfrac{f}{{d' + f}}\\
\Rightarrow \dfrac{h}{{h'}} = \dfrac{d}{{d'}} = \dfrac{f}{{d' + f}}\\
\Rightarrow d.d' + df = d'f\\
\Rightarrow \dfrac{1}{f} = - \dfrac{1}{{d'}} + \dfrac{1}{d}\\
\Rightarrow d' = \dfrac{{df}}{{f - d}}\\
\Rightarrow \dfrac{h}{{h'}} = \dfrac{d}{{d'}} = \dfrac{d}{{\dfrac{{df}}{{f - d}}}} = \dfrac{{f - d}}{f}
\end{array}\)
c) Ta có:
\(\begin{array}{l}
\dfrac{{AB}}{{A'B'}} = \dfrac{{OA}}{{OA'}} \Rightarrow \dfrac{h}{{h'}} = \dfrac{d}{{d'}}\\
\dfrac{{AB}}{{A'B'}} = \dfrac{{OI}}{{A'B'}} = \dfrac{{OF'}}{{ - OA' + OF}} = \dfrac{f}{{ - d' + f}}\\
\Rightarrow \dfrac{h}{{h'}} = \dfrac{d}{{d'}} = \dfrac{f}{{ - d' + f}}\\
\Rightarrow - d.d' + df = d'f\\
\Rightarrow - \dfrac{1}{f} = - \dfrac{1}{{d'}} + \dfrac{1}{d}\\
\Rightarrow d' = \dfrac{{df}}{{f + d}}\\
\Rightarrow \dfrac{h}{{h'}} = \dfrac{d}{{d'}} = \dfrac{d}{{\dfrac{{df}}{{f + d}}}} = \dfrac{{f + d}}{f}
\end{array}\)
