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A line L passes through the point $\mathrm{P}(5,-6,7)$ and is parallel to the planes $x+y+z=1$ and $2 x-y-2 z=3$. What are the direction ratios of the line of intersection of the given planes?
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Verified Answer
The correct answer is:
$ < 1,-4,3>$
Let $a, b, c$ be the direction ratios of the line Then its equation is
$\frac{x-5}{a}=\frac{y+6}{b}=\frac{z-7}{c}$
Since (i) is parallel to the planes $x+y+z=1$ and
$2 x-v-2 z=3$ then
$a(1)+b(1)+c(1)=0$ and $a(2)+b(-1)+c(-2)=0$
Bycross multiplication
$\frac{a}{-1}=\frac{b}{4}=\frac{c}{-3}=\lambda$
$\Rightarrow a=-\lambda, b=4 \lambda, c=-3 \lambda$
$\Rightarrow$ Direction ratios of the line are $ < -1,4,-3>= < 1,-4,3>$
$\frac{x-5}{a}=\frac{y+6}{b}=\frac{z-7}{c}$
Since (i) is parallel to the planes $x+y+z=1$ and
$2 x-v-2 z=3$ then
$a(1)+b(1)+c(1)=0$ and $a(2)+b(-1)+c(-2)=0$
Bycross multiplication
$\frac{a}{-1}=\frac{b}{4}=\frac{c}{-3}=\lambda$
$\Rightarrow a=-\lambda, b=4 \lambda, c=-3 \lambda$
$\Rightarrow$ Direction ratios of the line are $ < -1,4,-3>= < 1,-4,3>$
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