Cho a/x + b/y + c/z=0 x/a + y/b + z/c=1 chứng minh: x^2/a^2 + y^2/b^2 + z^2/c^2 =1

0 bình luận về “Cho a/x + b/y + c/z=0 x/a + y/b + z/c=1 chứng minh: x^2/a^2 + y^2/b^2 + z^2/c^2 =1”

1. Đáp án:

   Giải thích các bước giải:
   $\begin{array}{l}
   \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\\
   \Rightarrow {(\frac{x}{a} + \frac{y}{b} + \frac{z}{c})^2} = 1\\
   \Leftrightarrow \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} + 2 \times \frac{{xyc + yza + xzb}}{{abc}} = 1\\
   \\
   \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 0 \Leftrightarrow \frac{{ayz + xzb + xyc}}{{xyz}} = 0 \Leftrightarrow ayz + xzb + xyc = 0\\
   \Rightarrow \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1
   \end{array}$

   Trả lời

2. Ta có:

   `a/x+b/y+c/z=0`

   `⇒ayz+bxz+cxy=0`

   Ta có:

   ` x/a+y/b+z/c=1`

   `⇔(x/a+y/b+z/c)^2=1`

   `⇔x^2/a^2+y^2/b^2+z^2/c^2+2((xy)/(ab)+(yz)/(bc)+(zx)/(ca))=1`

   `⇔x^2/a^2+y^2/b^2+z^2/c^2+2((ayz+bxz+cxy)/(abc))=1`

   `⇔x^2/a^2+y^2/b^2+z^2/c^2+2.0=1`

   `⇔x^2/a^2+y^2/b^2+z^2/c^2=1`

   `⇒ĐPCM`

    

   Trả lời