Question

Cho \(\dfrac{a}{b+c}=\dfrac{b}{a+c}\) và (a ≠ +-b; a ≠ -c; b ≠ -c). Tính giá trị của M = \(\dfrac{c}{a+b}\)

Answer 1

Ta có:

\(\dfrac{a}{b+c}=\dfrac{b}{a+c}\)

\(\Rightarrow a\left(a+c\right)=b\left(b+c\right)\)

\(\Rightarrow a^2+ac=b^2+bc\)

\(\Rightarrow a^2-b^2=bc-ac\)

\(\Rightarrow\left(a+b\right)\left(a-b\right)=c\left(b-a\right)\)

\(\Rightarrow\left(a+b\right)\left(a-b\right)=-c\left(a-b\right)\)

\(\Rightarrow a+b=\dfrac{-c\left(a-b\right)}{a-b}\)

\(\Rightarrow a+b=-c\)

Thay \(a+b=-c\) ta có: 

\(M=\dfrac{c}{a+b}=\dfrac{c}{-c}=-1\)