x/a + y/b + z/c = 1 a/x + b/y + c/z = 0 tính x^2/a^2 + y^2/b^2 + z^2/c^2

0 bình luận về “x/a + y/b + z/c = 1 a/x + b/y + c/z = 0 tính x^2/a^2 + y^2/b^2 + z^2/c^2”

1. 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`

    

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2. Đáp án:

    1

   Giải thích các bước giải:

   \(\begin{array}{l}
   \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\\
    \Rightarrow {(\frac{x}{a} + \frac{y}{b} + \frac{z}{c})^2} = 1\\
    \Leftrightarrow \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} + \frac{{2xy}}{{ab}} + \frac{{2yz}}{{bc}} + \frac{{2xz}}{{ac}} = 1\\
    \Leftrightarrow \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} + \frac{{2(xyc + yza + xzb)}}{{abc}} = 1\\
   \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 0\\
    \Leftrightarrow \frac{{yza + xzb + xyc}}{{xyz}} = 0\\
    \Rightarrow yza + xzb + xyc = 0\\
    \Rightarrow \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} + \frac{0}{{abc}} = 1\\
    \Rightarrow \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1
   \end{array}\)

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