جواب فعالیت 2 صفحه 48 درس 2 هندسه دوازدهم (آشنایی با مقاطع مخروطی)

الف با توجه به قسمت اول فعالیت قبل می توان نوشت:

\(\begin{array}{l}FA + F'A = l\\\\ \Rightarrow FA + FA + FF' = l\\\\ \Rightarrow FA + FA = l - FF'\\\\ \Rightarrow 2FA = l - FF'\end{array}\)

به طریق مشابه، با در نظر گرفتن نقطه A'  می توان نوشت: 2FA=1-FF' ؛ پس :

\(\begin{array}{l}FA' = F'A = \frac{1}{2}l\\\\ \Rightarrow FF' + F'A' = FF' + + FA\\\\ \Rightarrow FA = F'A'\\\\OF = OF'\\\\ \Rightarrow OF + FA = OF' + F'A'\\\\ \Rightarrow OA = O'A' = a\end{array}\)

ب

\(\begin{array}{l}AF + AF' = l\;\;\\\\\mathop \Rightarrow \limits^{AF = A'F'} \;\;A'F' + AF' = l\\\\ \Rightarrow AA' = l \Rightarrow l = 2a\end{array}\)

الف$\begin{array}{l}\end{array}$

$\begin{array}{l}\end{array}$\(\begin{array}{l}\Delta OBF \cong \Delta OBF'\\\\ \Rightarrow BF = BF'\\\\BF + BF' = AA' = 2a\\\\ \Rightarrow BF + BF = 2a\\\\ \Rightarrow 2BF = 2a \Rightarrow BF = a\\\\\Delta OBF:\widehat O = {90^ \circ }\\\\ \Rightarrow O{B^2} + O{F^2} = B{F^2}\\\\ \Rightarrow O{B^2} + {c^2} = {a^2}\quad (1)\end{array}\)

ب

\(\begin{array}{l}\Delta OB'F \cong \Delta OB'F'\\\\ \Rightarrow B'F = B'F'\\\\B'F + B'F' = AA' = 2a\\\\ \Rightarrow 2BF = 2a \Rightarrow BF = a\\\\\Delta OBF:\widehat O = {90^ \circ }\\\\ \Rightarrow O{{B'}^2} + O{F^2} = B'{F^2}\\\\ \Rightarrow O{{B'}^2} + {c^2} = {a^2}\quad (2)\\\\(1)\;,\;(2) \Rightarrow O{B^2} = O{{B'}^2}\\\\ \Rightarrow OB = OB' = b\\\\ \Rightarrow {a^2} = {b^2} + {c^2}\end{array}\)