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Mathematics · 2025 · 4 marks
CBSE 2025 · Region 4 · Set 2 · Q37
An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line A and Line B, each consisting of multiple stations are…
An engineer is designing a new metro rail network in a city.
Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by $\displaystyle l_{1}: \frac{\mathrm{x}-2}{3}=\frac{\mathrm{y}+1}{-2}=\frac{\mathrm{z}-3}{4}$, while the track for Line B is represented by $\displaystyle l_{2}: \frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}-3}{1}=\frac{\mathrm{z}+2}{-3}$. Based on the above information, answer the following questions :
(i)
Find whether the two metro tracks are parallel.
(ii)
Solar panels are to be installed on the rooftop of the metro stations. Determine the equation of the line representing the placement of solar panels on the rooftop of Line A's stations, given that panels are to be positioned parallel to Line A's track $\displaystyle \left(l_{1}\right)$ and pass through the point $\displaystyle (1,-2,-3)$.
(iii)(b)
Find the shortest distance between Line A and Line B . Case Study - $\displaystyle 3$
Marking-scheme solution
\[l_{1}: \frac{\mathrm{x}-2}{3}=\frac{\mathrm{y}+1}{-2}=\frac{\mathrm{z}-3}{4} ; l_{2}: \frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}-3}{1}=\frac{\mathrm{z}+2}{-3}
\]
(i)
Drsof $\displaystyle l_{1}$ are $\displaystyle <3,-2,4>, \operatorname{drs}$ of $\displaystyle l_{2}$ are $\displaystyle <2,1,-3>$ as drs are not proportional, hence $\displaystyle l_{1}$ is not parallel to $\displaystyle l_{2}$.
(ii)
Equations of line parallel to $\displaystyle l_{1}$ and passing through( $\displaystyle 1,-2,-3$ )is
\[\frac{\mathrm{x}-1}{3}=\frac{\mathrm{y}+2}{-2}=\frac{\mathrm{z}+3}{4} \text { or } \vec{r}=(\hat{i}-2 \hat{j}-3 \hat{k})+\lambda(3 \hat{i}-2 \hat{j}+4 \hat{k})
\]
(iii)(a)
The pathway is perpendicular to $\displaystyle l_{1}$ and $\displaystyle l_{2} .$. It is parallel to $\displaystyle \vec{b}_{1} \times \vec{b}_{2}$
\[\vec{b}=\vec{b}_{1} \times \vec{b}_{2}=\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
3 & -2 & 4 \\
2 & 1 & -3
\end{array}\right|=2 \hat{i}+17 \hat{j}+7 \hat{k}
\]
∴ Equation of pathwayis $\displaystyle \vec{r}=(3 \hat{i}+2 \hat{j}+\hat{k})+\lambda(2 \hat{i}+17 \hat{j}+7 \hat{k})$
(iii)
$\displaystyle (b) \vec{a}_{1}=2 \hat{i}-\hat{j}+3 \hat{k}, \vec{a}_{2}=\hat{i}+3 \hat{j}-2 \hat{k}$
\[\begin{aligned}
\vec{b}_{1} & =3 \hat{i}-2 \hat{j}+4 \hat{k}, \vec{b}_{2}=2 \hat{i}+\hat{j}-3 \hat{k} \\
d & =\left|\frac{\left(\vec{a}_{2}-\vec{a}_{1}\right) \cdot\left(\vec{b}_{1} \times \vec{b}_{2}\right)}{\left|\vec{b}_{1} \times \vec{b}_{2}\right|}\right| \\
& =\left|\frac{(-\hat{i}+4 \hat{j}-5 \hat{k}) \cdot(2 \hat{i}+17 \hat{j}+7 \hat{k})}{\sqrt{4+289+49}}\right|
\end{aligned}
\]
\[=\frac{31}{\sqrt{342}}
\]
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