Question

مطلوب است محاسبه:$؟= \frac{1}{b} +\frac{1}{a} $در صورتی که: $\frac{b}{a} + \frac{a}{b}=5 $و $ \frac{ b^{2} }{a} + \frac{ a^{2} }{b} =12$

Answer 1

$$ \frac{a}{b} + \frac{b}{a}=5 \Longrightarrow a^{2} + b^{2} =5ab ;; \frac{ a^{2} }{b} + \frac{ b^{2} }{a} =12 \Longrightarrow a^{3} + b^{3} =12ab;; \Longrightarrow (a+b)( a^{2} -ab+ b^{2} )=12ab;;(a+b)(4ab)=12ab \Longrightarrow a+b=3 \Longrightarrow a^{2} +2ab+ b^{2} =9 \Longrightarrow ab= \frac{9}{7} \Longrightarrow \frac{1}{a} + \frac{1}{b}= \frac{a+b}{ab} \Longrightarrow \frac{1}{a} + \frac{1}{b} = \frac{3}{ \frac{9}{7} } = \frac{7}{3} $$