Question

CM nếu (a^2+b^2) .(x^2+y^2)=(ax+by)^2 thì ay-bx=0

Answer 1

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

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

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

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

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

\(\Leftrightarrow a^2x^2+b^2x^2=2abxy\)

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

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

\(\Leftrightarrow ax-bx=0\left(đpcm\right)\)