Chứng minh rằng nếu (a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = (ax + by + cz)^2 với x, y, z khác 0 thì a/x = b/y = c/z
Chứng minh rằng nếu (a2 + b2 + c2)(x2 + y2 + z2) = (ax + by + cz)2 với x, y, z khác 0 thì \(\frac{a}{x} = \frac{b}{y} = \frac{c}{z}\).
Lời giải
Ta có: (a2 + b2 + c2)(x2 + y2 + z2) = (ax + by + cz)2
⇔ a2x2 + a2y2 + a2z2 + b2y2 + b2z2 + c2x2 + c2y2 + c2z2 = a2x2 + b2y2 + c2z2 + 2axby + 2axcz + 2bycz
⇔ a2y2 + a2z2 + b2x2 + b2z2 + c2x2 + c2y2 – 2axby – 2axcz – 2bycz = 0
⇔ (a2y2 – 2axby + b2x2) + (a2z2 – 2axcz + c2x2) + (b2z2 – 2bycz + c2y2) = 0
⇔ (ay – by)2 + (az – cx)2 + (bz – cy)2 = 0
Vì (ay – bx)2 ≥ 0; (az – cx)2 ≥ 0; (bz – cy)2 ≥ 0 nên
(ay – by)2 + (az – cx)2 + (bz – cy)2 ≥ 0
Vậy dấu “=” xảy ra khi:
\(\left\{ \begin{array}{l}ay = bx\\az = cx\\bz = cy\end{array} \right. \Rightarrow \left\{ \begin{array}{l}\frac{a}{x} = \frac{b}{y}\\\frac{a}{x} = \frac{c}{z}\\\frac{b}{y} = \frac{c}{z}\end{array} \right.\)
\( \Rightarrow \frac{a}{x} = \frac{b}{y} = \frac{c}{z}\) (xyz ≠ 0). (đpcm)