Question

Cho \(\frac{1}{c}\)= \(\frac{1}{2}\)\(\left(\frac{1}{a}+\frac{1}{b}\right)\)(với a,b,c khác 0;b khác c).Chứng minh rằng\(\frac{a}{b}\)=\(\frac{a-c}{c-b}\)

Answer 1

Từ \(gt\Leftrightarrow\frac{1}{c}=\frac{1}{2}.\frac{a+b}{ab}\)

\(\Leftrightarrow\frac{1}{c}=\frac{a+b}{2ab}\Leftrightarrow c\left(a+b\right)=2ab\Leftrightarrow ac+bc=ab+ab\)

\(\Leftrightarrow ac-ab=ab-bc\Leftrightarrow a\left(c-b\right)=b\left(a-c\right)\)

\(\Rightarrow\frac{a}{b}=\frac{a-c}{b-c}\) (đpcm)

Answer 2

Từ \(gt\Leftrightarrow\frac{1}{c}=\frac{1}{2}.\frac{a+b}{ab}\)

\(\Leftrightarrow\frac{1}{c}=\frac{a+b}{2ab}\Leftrightarrow c\left(a+b\right)=2ab\Leftrightarrow ac+bc=ab+ab\)

\(\Leftrightarrow ac-ab=ab-bc\Leftrightarrow a\left(c-b\right)=b\left(a-c\right)\)

\(\Rightarrow\frac{a}{b}=\frac{a-c}{b-c}\)

\(\Rightarrowđpcm\)