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Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry5 Marks
Question
Determine whether the following pair of lines intersect or not:
$\frac{\text{x}-1}{2}=\frac{\text{y}+1}{3}=\text{z}$ and $\frac{\text{x}+1}{5}=\frac{\text{y}-2}{1};\text{z}=2$
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Answer

Given, equation of first line is$\frac{\text{x}-1}{2}=\frac{\text{y}+1}{3}=\frac{\text{}z}{1}=\lambda\text{ (say) }\dots(1)$
General point on line (1) is
$\big(2\lambda+1,3\lambda-1,\lambda\big)$
Another equation of line is
$\frac{\text{x}-1}{5}=\frac{\text{y}-2}{1},\text{z}=3\dots(2)$
$\frac{\text{x}-5}{5}=\frac{\text{y}-2}{1}=\mu,\text{ (say) },\text{z}=3$
General point on line (2) is
$\big(5\mu+1,\mu+2,3\big)$
If line (1) and (2) intersect each other then, there is a common point to them, so, we must have of $\lambda$ and $\mu$ such that
$2\lambda+1=5\mu+1\Rightarrow2\lambda-5\mu=0\dots(3)$
$3\lambda-1=\mu+2\Rightarrow3\lambda-\mu=3\dots(4)$
$\lambda=3\Rightarrow\lambda=3\dots(5)$
Put value of $\lambda$ in equation (4),
$3\lambda-\mu=3$
$3(3)-\mu=3$
$-\mu=3-9$
$\mu=6$
Put the value of $\lambda$ and $\mu$ in equation (3), so
$2\lambda-5\mu=0$
$2(3)-5(6)=0$
$6-30=0$
$-24\neq0$
$\text{LHS}\neq\text{RHS}$
sice the values of $\lambda$ and $\mu$ obtained from equation (4) and (5) dose not satisfy equation (3),
So, given lines are not intersecting.

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