Question

Cho \(\frac{a}{b}=\frac{3}{b}=\frac{b}{a}\). Chứng minh rằng a = b (với a+b không bằng -3)

 

Answer 1

áp dụng t/c dãy tỉ số bằng nhau ta có:

\(\frac{a}{b}=\frac{3}{b}=\frac{b}{a}\Rightarrow\frac{a+3+b}{2b+a}\)

\(\frac{3}{b}=\frac{a+3+b}{2b+a}\Rightarrow3.\left(2b+a\right)=b.\left(a+3+b\right)\)

\(\Rightarrow6b+3a=ab+3b+b^2\)

\(3b+3a=ab+b^2\)

\(3.\left(a+b\right)=b.\left(a+b\right)\)

\(\Rightarrow b=3\)

\(\frac{a}{b}=\frac{a+b+3}{2b+a}\)

\(\Rightarrow2ab+a^2=ab+b^2+3b\)

\(ab+a^2=b^2+3b\Rightarrow3.a+a^2=3.b+b^2\)

\(\Rightarrow a=b\left(đpcm\right)\)