Question

cho a+b+c=1/2019; ax+by+cz=0. c/m:ax^2+by^2+cz^2/bc(y-z)^2+ca(z-x)^2+ab(x-y)^2= 2019

giúp mik với

Answer 1

Lời giải:

Từ \(ax+by+cz=0\Rightarrow (ax+by+cz)^2=0\)

\(\Leftrightarrow a^2x^2+b^2y^2+c^2z^2=-2(axby+axcz+bycz)\)

\(=-2(bcyz+cazx+abxy)\)

Khi đó:

\(bc(y-z)^2+ca(z-x)^2+ab(x-y)^2=bc(y^2-2yz+z^2)+ca(z^2-2zx+x^2)+ab(x^2-2xy+y^2)\)

\(=(bcy^2+bcz^2+caz^2+cax^2+abx^2+aby^2)-(2bcyz+2cazx+2abxy)\)

\(=(bcy^2+bcz^2+caz^2+cax^2+abx^2+aby^2)+(a^2x^2+b^2y^2+c^2z^2)\)

\(=ax^2(a+b+c)+by^2(a+b+c)+cz^2(a+b+c)=(a+b+c)(ax^2+by^2+cz^2)\)

Do đó:

\(\frac{ax^2+by^2+cz^2}{bc(y-z)^2+ca(z-x)^2+ab(x-y)^2}=\frac{ax^2+by^2+cz^2}{(ax^2+by^2+c^2)(a+b+c)}=\frac{1}{a+b+c}=\frac{1}{\frac{1}{2019}}=2019\)

Ta có đpcm.