Question

Cho tỉ lệ thức \(\dfrac{a}{b}\)=\(\dfrac{c}{d}\)

CMR:\(\dfrac{ac}{bd}\)=\(\dfrac{2a^2+3c^2}{2b^2+3d^2}\)

Answer 1

Đặt :

\(\dfrac{a}{b}=\dfrac{c}{d}=k\) \(\Leftrightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\)

\(\dfrac{ac}{bd}=\dfrac{bk.dk}{bd}=k^2\left(1\right)\)

\(\dfrac{2a^2+3c^2}{2b^2+3d^2}=\dfrac{2.\left(bk\right)^2+3.\left(dk\right)^2}{2b^2+3d^2}=\dfrac{k^2\left(2b^2+3d^2\right)}{3b^2+3d^2}=k^2\left(2\right)\)

Từ \(\left(1\right)+\left(2\right)\Leftrightarrow\dfrac{ac}{bd}=\dfrac{2a^2+3c^2}{2b^2+3d^2}\left(đpcm\right)\)