Question

Cho tỉ lệ thức \(\dfrac{a^2+b^2}{c^2+d^2}\)=\(\dfrac{ab}{cd}\)với a,b,c,d ≠0 và c≠-d. CM \(\dfrac{a}{b}\)=\(\dfrac{c}{d}\) hoặc \(\dfrac{a}{b}\)=\(\dfrac{d}{c}\)mn giúp mình vs ạ

Answer 1

\(\dfrac{a^2+b^2}{c^2+d^2}=\dfrac{ab}{cd}\)

=>\(cd\left(a^2+b^2\right)=ab\left(c^2+d^2\right)\)

=>\(cda^2+cdb^2=abc^2+abd^2\)

=>\(cda^2-abc^2=abd^2-cdb^2\)

=>\(ac\left(da-bc\right)=bd\left(ad-cb\right)\)

=>\(\left(ac-bd\right)\left(ad-bc\right)=0\)

=>\(\left[{}\begin{matrix}ac=bd\\ad=bc\end{matrix}\right.\)

=>\(\left[{}\begin{matrix}\dfrac{a}{b}=\dfrac{d}{c}\\\dfrac{a}{b}=\dfrac{c}{d}\end{matrix}\right.\)