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Straight line in space — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsStraight line in space5 Marks
Question
Determine whether the following pair of lines intersect or not:
$\frac{\text{x}-1}{3}=\frac{\text{y}-1}{-1}=\frac{\text{z}+1}{0}$ and $\frac{\text{x}-4}{2}=\frac{\text{y}-0}{0}=\frac{\text{z}+1}{3}$
✓

Answer

Given, equation of first line is,
$\frac{\text{x}-1}{3}=\frac{\text{y}-1}{-1}=\frac{\text{z}+1}{0}=\lambda\text{ (say) }\dots(1)$
General point on line (1) is,
$\big(3\lambda+1,-\lambda+1,-1\big)$
Another equation of line is
$\frac{\text{x}-4}{2}=\frac{\text{y}-0}{0}=\frac{\text{z}+1}{3}=\mu\text{ (say) }\dots(2)$
General point on line (2) is,
$\big(2\mu+4,0,3\mu-1\big)$
If line (1) and (2) intersecting then there must be a common point, so, we must have the value of $\lambda$ and $\mu$ as
$3\lambda+1=2\mu+4\Rightarrow3\lambda-2\mu=3\dots(1)$
$=\lambda+1=0\Rightarrow\lambda=1\dots(2)$
$3\mu-1=-1\Rightarrow\mu=0\dots(3)$
Put the value of $\lambda$ and $\mu$ in equation (1), so
$3\lambda-2\mu=3$
$3(1)-2(0)=3$
$3=3$
$\text{LHS}\neq\text{RHS}$
Since the values of $\lambda$ and $\mu$ obtained by equation (2) and (3) satisfy equation (1), so,
given lines are intersecting.

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