Question

Cho x/a + y/b + z/c =1 và a/x + b/y + c/z = 0 .

Chưng minh rằng x^2/a^2 + y^2/b^2 + z^2/c^2 =1

Answer 1

Ta có :
+ ) \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0\)

\(\Leftrightarrow\frac{ayz+bxz+cxy}{xyz}=0\)

\(\Leftrightarrow ayz+bxz+cxy=0\)

+ ) \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\)

\(\Leftrightarrow\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=1\)

\(\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{xy}{ab}+\frac{yz}{bc}+\frac{xz}{zc}\right)=1\)

\(\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{ayz+bxz+cxy}{abc}\right)=1\)

\(\Leftrightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\left(đpcm\right)\)

Chúc bạn học tốt !!!