Question

Cho tỉ lệ thức \(\dfrac{a}{c}\)= \(\dfrac{c}{b}\). Chứng minh tỉ lệ thức \(\dfrac{2a^2+c^2}{b^2+2c^2}\)

Answer 1

Sửa đề: Chứng minh \(\frac{2a^2+c^2}{b^2+2c^2}=\frac{a}{b}\)

Đặt \(\frac{a}{c}=\frac{c}{b}=k\)

=>\(\begin{cases}c=bk\\ a=ck=bk\cdot k=bk^2\end{cases}\)

\(\frac{2a^2+c^2}{b^2+2c^2}=\frac{2\cdot\left(bk^2\right)^2+\left(bk\right)^2}{b^2+2\cdot\left(bk\right)^2}=\frac{2b^2k^4+b^2k^2}{b^2+2b^2k^2}=\frac{b^2k^2\left(2k^2+1\right)}{b^2\left(1+2k^2\right)}=k^2\)

\(\frac{a}{b}=\frac{bk^2}{b}=k^2\)

Do đó; \(\frac{2a^2+c^2}{b^2+2c^2}=\frac{a}{b}\)