Question

Chứng minh rằng nếu (a²+b²)(x²+y²) = (ax+by)² với x,y khác 0 thì \(\dfrac{a}{x}=\dfrac{b}{y}\)

Answer 1

Ta có:

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

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

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

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

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

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

\(\Leftrightarrow\) \(ay=bx\)

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