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Straight line in space — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsStraight line in space5 Marks
Question
Determine whether the following pair of lines intersect or not:
$\frac{\text{x}-5}{4}=\frac{\text{y}-7}{4}=\frac{\text{z}+3}{-5}$ and $\frac{\text{x}-8}{7}=\frac{\text{y}-4}{1}=\frac{3-5}{3}$
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Answer

Given, equation of line is $\frac{\text{x}-5}{4}=\frac{\text{y}-7}{4}=\frac{\text{z}+3}{-5}=\lambda\text{ (say) }\dots(1)$ General point on line (1) is, $\big(4\lambda+5,4\lambda+7,-5\lambda-3\big)$ Another equation of line is, $\frac{\text{x}-8}{7}=\frac{\text{y}-4}{1}=\frac{3-5}{3}=\mu\text{ (say) }\dots(2)$ General point on line (2) is $\big(7\mu+8,\mu+4,3\mu+5\big)$ If line (1) and (2) intersecting, then there must have some common point to them, so, we must have value of $\lambda$ and $\mu$ such that $4\lambda+5=7\mu+8\Rightarrow4\lambda-7\mu=3\dots(3)$ $4\lambda+5=\mu+4\Rightarrow4\lambda-\mu=-3\dots(4)$ $-5\lambda-3=3\mu+5\Rightarrow-5\lambda-3\mu=8\dots(5)$ Solving equation (3) and (4) to find $\lambda$ and $\mu,$
$\mu=-1$ Put value of $\lambda$ in equation (3), $4\lambda-7\mu=3$ $4\lambda-7(-1)=3$ $4\lambda=3-7$ $\lambda=-1$ Put the value of $\lambda$ and $\mu$ in equation (5), $-5\lambda-3\mu=8$ $-5(-1)-3(-1)=8$ $5+3=8$ $\text{LHS = RHS}$

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