Question

Cho 2 số hữu tỉ a và b thỏa a+b=ab=a/b :

1. Chứng minh a/b =a-1

2. Chứng minh b=-1

3. Tìm a

Answer 1

\(a+b=ab=\dfrac{a}{b}\)

Ta có:

\(ab=\dfrac{a}{b}\Rightarrow ab=\dfrac{a^2}{ab}\)

\(\Rightarrow a^2b^2=a^2\)

\(\Rightarrow b^2=1\Rightarrow b=\pm1\)

Xét:

\(b=1\Rightarrow a+b=ab=\dfrac{a}{b}\Rightarrow a+1=a=a\left(KTM\right)\)

Xét:

\(b=-1\Rightarrow a+b=ab=\dfrac{a}{b}\Rightarrow a-1=-a=-a\)

\(\Rightarrow a-1=-a\)

\(\Rightarrow2a=1\Rightarrow a=\dfrac{1}{2}\)

Ta có:

\(\dfrac{a}{b}=a-1\rightarrowđpcm\)

\(b=-1\rightarrowđpcm\)

\(a=\dfrac{1}{2}\)