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If the direction ratios $a, b, c$ of a line $L$ satisfy the relations $a b+b c+c a=0$ and $6 a b+9 b c+8 c a=0$, then the direction cosines of the line $L$ are
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$\frac{-3}{7}, \frac{2}{7}, \frac{-6}{7}$
Given relations $a b+b c+c a=0$
and $\quad 6 a b+9 b c+8 c a=0$
$\therefore \quad 3 b c+2 a c=0 \Rightarrow 3 b+2 a=0$
and $\quad-2 a b+b c=0 \Rightarrow 2 a-c=0$
$\therefore \quad 2 a=-3 b=c \Rightarrow \frac{a}{-3}=\frac{b}{2}=\frac{c}{-6}$
or $\quad \frac{a}{-3 / 7}=\frac{b}{2 / 7}=\frac{c}{-6 / 7}$
$\therefore$ Direction cosines of line $L$ is $\frac{-3}{7}, \frac{2}{7}, \frac{-6}{7}$
and $\quad 6 a b+9 b c+8 c a=0$
$\therefore \quad 3 b c+2 a c=0 \Rightarrow 3 b+2 a=0$
and $\quad-2 a b+b c=0 \Rightarrow 2 a-c=0$
$\therefore \quad 2 a=-3 b=c \Rightarrow \frac{a}{-3}=\frac{b}{2}=\frac{c}{-6}$
or $\quad \frac{a}{-3 / 7}=\frac{b}{2 / 7}=\frac{c}{-6 / 7}$
$\therefore$ Direction cosines of line $L$ is $\frac{-3}{7}, \frac{2}{7}, \frac{-6}{7}$
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