Question

[Cô nàng Bạch Dương]

CMR: Nếu (a² + b²).(x² + y²) =(ax + by)²  với mọi x,y thì \(\frac{a}{x}\)^{=\(\frac{b}{y}\)}

Answer 1

[Lê Tài Bảo Châu]

\(\left(a^2+b^2\right)\left(x^2+y^2\right)=\left(ax+by\right)^2\)

\(\Leftrightarrow a^2x^2+b^2x^2+a^2y^2+b^2y^2=a^2x^2+2axby+b^2y^2\)

\(\Leftrightarrow a^2x^2+b^2x^2+a^2y^2+b^2y^2-a^2x^2-2axby-b^2y^2=0\)

\(\Leftrightarrow a^2y^2-2axby+b^2x^2=0\)

\(\Leftrightarrow a^2b^2-axby-axby+b^2x^2=0\)

\(\Leftrightarrow ay\left(ay-bx\right)-bx\left(ay-bx\right)=0\)

\(\Leftrightarrow\left(ay-bx\right)\left(ay-bx\right)=0\)

\(\Leftrightarrow\left(ay-bx\right)^2=0\)

\(\Leftrightarrow ay-bx=0\)

\(\Leftrightarrow ay=bx\)

\(\Leftrightarrow\frac{a}{x}=\frac{b}{y}\)