Question

Cho ab/bc=b/c.Chứng minh a^2+b^2/c^2+d^2=a/c

Answer 1

Sửa đề: Chứng minh \(\frac{a^2+b^2}{c^2+b^2}=\frac{a}{c}\)

Ta có: \(\frac{\overline{ab}}{\overline{bc}}=\frac{b}{c}\)

=>\(\frac{10a+b}{10b+c}=\frac{b}{c}\)

=>c(10a+b)=b(10b+c)

=>10ac+bc=10b^2+bc

=>\(10ac=10b^2\)

=>\(ac=b^2\)

=>\(\frac{a}{b}=\frac{b}{c}\)

Đặt \(\frac{a}{b}=\frac{b}{c}=k\)

=>\(\begin{cases}b=ck\\ a=bk=ck\cdot k=ck^2\end{cases}\)

\(\frac{a^2+b^2}{c^2+b^2}=\frac{\left(ck^2\right)^2+\left(ck^{}\right)^2}{c^2+\left(ck\right)^2}=\frac{c^2k^4+c^2k^2}{c^2+c^2k^2}=\frac{c^2k^2\left(k^2+1\right)}{c^2\left(1+k^2\right)}=k^2\)

\(\frac{a}{c}=\frac{ck^2}{c}=k^2\)

Do đó: \(\frac{a^2+b^2}{c^2+b^2}=\frac{a}{c}\)