Question

Cho tỉ lệ thức : a/b=c/d

CM : a*c/b*d=a^2-c^2/b^2-d^2

Answer 1

Đặt \(\dfrac{a}{b}=\dfrac{c}{d}=k\)

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

Ta có: \(\dfrac{a\cdot c}{b\cdot d}=\dfrac{bk\cdot dk}{b\cdot d}=k^2\)

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

Do đó: \(\dfrac{ac}{bd}=\dfrac{a^2-c^2}{b^2-d^2}\)