Question

Cho a/b=c/d.Chứng minh rằng:a^2-b^2/c^2-d^2=ab/cd

Answer 1

Đặt \(\frac{a}{b}=\frac{c}{d}=k\Leftrightarrow a=bk;c=dk\)

Thay a = bk, c = dk vào \(\frac{a^2-b^2}{c^2-d^2}\) và \(\frac{ab}{cd}\), ta có:

\(\frac{a^2-b^2}{c^2-d^2}=\frac{\left(bk\right)^2-b^2}{\left(dk\right)^2-d^2}=\frac{b^2k^2-b^2}{d^2k^2-d^2}=\frac{b^2\left(k^2-1\right)}{d^2\left(k^2-1\right)}=\frac{b^2}{d^2}\)

\(\frac{ab}{cd}=\frac{bk.b}{dk.d}=\frac{b^2.k}{d^2.k}=\frac{b^2}{d^2}\)

\(\Rightarrow\frac{a^2-b^2}{c^2-d^2}=\frac{ab}{cd}\left(đpcm\right)\)