If $a x_{1}^{2} + b y_{1}^{2} + c z_{1}^{2} = a x_{2}^{2} + b y_{2}^{2} + c z_{2}^{2} = a x_{3}^{2} + b y_{3}^{2} + c z^{3} = d, a x_{2} x_{3} + b y_{2} y_{3} + c z_{2} z_{3} = a x_{3} x_{1} + b y_{3} y_{1} + c z_{3} z_{1} = a x_{1} x_{2} + b y_{1} y_{2} + c z_{1} z_{2} = f$ , then $∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & z_{1} x_{2} & y_{2} & z_{2} x_{3} & y_{3} & z_{3} \end{matrix} ∣ ∣ ∣ ∣ = ?$

(d - f) {\frac{d + 2 f)}{a b c}}^{\frac{1}{2}}

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- (d - f) {\frac{d + 2 f)}{a b c}}^{\frac{1}{2}}

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(d - f) {\frac{d + 2 f)}{a b c}}^{\frac{- 1}{2}}

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- (d - f) {\frac{d + 2 f)}{a b c}}^{\frac{- 1}{2}}

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Solution

The correct option is A $(d - f) {\frac{d + 2 f)}{a b c}}^{\frac{1}{2}}$
Let $D = ∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & z_{1} x_{2} & y_{2} & z_{2} x_{3} & y_{3} & z_{3} \end{matrix} ∣ ∣ ∣ ∣$
$∴ D^{2} = D \times D = ∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & z_{1} x_{2} & y_{2} & z_{2} x_{3} & y_{3} & z_{3} \end{matrix} ∣ ∣ ∣ ∣ \times ∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & z_{1} x_{2} & y_{2} & z_{2} x_{3} & y_{3} & z_{3} \end{matrix} ∣ ∣ ∣ ∣$
$= \frac{1}{a b c} ∣ ∣ ∣ ∣ \begin{matrix} x_{1} & y_{1} & z_{1} x_{2} & y_{2} & z_{2} x_{3} & y_{3} & z_{3} \end{matrix} ∣ ∣ ∣ ∣ \times ∣ ∣ ∣ ∣ \begin{matrix} a x_{1} & b y_{1} & c z_{1} a x_{2} & b y_{2} & c z_{2} a x_{3} & b y_{3} & c z_{3} \end{matrix} ∣ ∣ ∣ ∣$
$∣ ∣ ∣ ∣ ∣ \begin{matrix} a x_{1}^{2} + b y_{1}^{2} + c z_{1}^{2} & a x_{1} x_{2} + b y_{1} y_{2} + c z_{1} z_{2} & a x_{3} x_{1} + b y_{3} y_{1} a x_{1} x_{2} + b y_{1} y_{2} + c z_{1} z_{2} & a x_{2}^{2} + b y_{2}^{2} + c z_{2}^{2} & a x_{2} x_{3} + b y_{2} y_{3} + c z_{2} z_{3} a x_{3} x_{1} + b y_{3} y_{1} + c z_{3} z_{1} & a x_{2} x_{3} + b y_{2} y_{3} + c z_{2} z_{3} & a x_{3}^{2} + b y_{3}^{2} + c z_{3}^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$ (Multiplying Row by Row)
$= \frac{1}{a b c} ∣ ∣ ∣ ∣ \begin{matrix} d & f & f f & d & f f & f & d \end{matrix} ∣ ∣ ∣ ∣$
$= \frac{1}{a b c} ∣ ∣ ∣ ∣ \begin{matrix} d + 2 f & f & f d + 2 f & d & f d + 2 f & f & d \end{matrix} ∣ ∣ ∣ ∣$ ------ $[C_{1} \to C_{1} + C_{2} + C_{3}]$

$= \frac{(d + 2 f)}{a b c} ∣ ∣ ∣ ∣ \begin{matrix} 1 & f & f 1 & d & f 1 & f & d \end{matrix} ∣ ∣ ∣ ∣$
$= \frac{(d + 2 f)}{a b c} ∣ ∣ ∣ ∣ \begin{matrix} 1 & f & f 0 & d - f & 0 1 & f & d \end{matrix} ∣ ∣ ∣ ∣ . . . . . . . [R_{2} \to R_{2} - R_{1}]$
$= \frac{(d + 2 f)}{a b c} \cdot (d - f) ∣ ∣ ∣ \begin{matrix} 1 & f 1 & d \end{matrix} ∣ ∣ ∣$
$= \frac{(d + 2 f)}{a b c} \cdot (d - f) \cdot (d - f) = \frac{(d + 2 f) (d - f)^{2}}{a b c}$
$∴ D = (d - f) {\frac{d + 2 f}{a b c}}^{1 / 2}$

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Ifax21+by21+cz21=ax22+by22+cz22=ax23+by23+cz3=d,ax2x3+by2y3+cz2z3=ax3x1+by3y1+cz3z1=ax1x2+by1y2+cz1z2=f, then ∣∣ ∣∣x1y1z1x2y2z2x3y3z3∣∣ ∣∣=?