Question

cho a,b,c,x,y,z>0

CMR \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\)⇔\(\sqrt{ax}+\sqrt{by}+\sqrt{cz}=\sqrt{\left(a+b+c\right)\left(x+y+z\right)}\)

giúp mình với mình cần gấp lắm =))))))

Answer 1

\(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\)⇒ \(ay=bx;bz=cy;cx=az\)

⇒ \(\left(\sqrt{ay}-\sqrt{bx}\right)^2+\left(\sqrt{bz}-\sqrt{cy}\right)^2+\left(\sqrt{cx}-\sqrt{az}\right)^2\)\(=0\)

⇒ \(ay+az+bx+bz+cx+cy=2\left(\sqrt{aybx}+\sqrt{bzcy}+\sqrt{cxaz}\right)\)

⇒ \(ax+ay+az+bx+by+bz+cx+cy+cz=ax+by+cz+2\left(\sqrt{axby}+\sqrt{bycz}+\sqrt{czax}\right)\)

⇒ \(\left(a+b+c\right)\left(x+y+z\right)=\left(\sqrt{ax}+\sqrt{by}+\sqrt{cz}\right)^2\)

⇒ \(\sqrt{ax}+\sqrt{by}+\sqrt{cz}=\sqrt{\left(a+b+c\right)\left(x+y+z\right)}\)

Vậy ....