Question

Cho \(\dfrac{a}{b}=\dfrac{c}{d}\) CMR :
\(\dfrac{ab}{cd}=\dfrac{a^2-b^2}{c^2-d^2}\)

Answer 1

Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)

\(\Rightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\)

Suy ra:

+ \(\dfrac{ab}{cd}=\dfrac{bkb}{dkd}=\dfrac{kb^2}{kd^2}=\dfrac{b^2}{d^2}\)

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

=> đpcm