Question

Cho \(\frac{a}{c}\)= \(\frac{c}{b}\). Chứng minh rằng \(\frac{a^2+c^2}{b^2+c^2}\)= \(\frac{a}{b}\)

 

 

 

Answer 1

Đặt \(\frac{a}{c}=\frac{c}{b}=k\Rightarrow\begin{cases}a=ck\\c=bk\Rightarrow b=\frac{c}{k}\end{cases}\)

\(VT=\frac{c^2k^2+c^2}{\frac{c^2}{k^2}+c^2}=k^2\)

\(VP=\frac{a}{b}=\frac{ck}{\frac{c}{k}}=k^2\)

=> VT=VP (dpcm)

Answer 2

theo đề bài ta có: 

\(\frac{a}{c}=\frac{c}{b}\) => a.b=c²

khi đó :

 \(\frac{a^2+c^2}{b^2+c^2}=\frac{a^2+ab}{b^2+ab}=\frac{a\left(a+b\right)}{b\left(a+b\right)}=\frac{a}{b}\)

vậy khi \(\frac{a}{c}=\frac{c}{b}\) thì \(\frac{a^2+c^2}{b^2+c^2}=\frac{a}{b}\)