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Mathematics · 2022 · 4 marks
CBSE 2022 · Region 1 · Set 2 · Q14
Electrical transmission wires which are laid down in winters are stretched tightly to accommodate expansion in summers. Two such wires lie along the…
Electrical transmission wires which are laid down in winters are stretched tightly to accommodate expansion in summers.
Two such wires lie along the following lines : $\displaystyle l_{1}: \frac{x+1}{3}=\frac{\mathrm{y}-3}{-2}=\frac{\mathrm{z}+2}{-1}$ $\displaystyle l_{2}: \frac{x}{-1}=\frac{\mathrm{y}-7}{3}=\frac{\mathrm{z}+7}{-2}$ Based on the given information, answer the following questions :
(i)
Are the lines $\displaystyle l_{1}$ and $\displaystyle l_{2}$ coplanar ? Justify your answer.
(ii)
Find the point of intersection of the lines $\displaystyle l_{1}$ and $\displaystyle l_{2}$.
Marking-scheme solution
(i)
\(\displaystyle \) Consider \(\left|\begin{array}{ccc}x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\ A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2}\end{array}\right|\)
\[\begin{aligned}
& =\left|\begin{array}{ccc}
+1 & 4 & -5 \\
3 & -2 & -1 \\
-1 & 3 & -2
\end{array}\right| \\
& =1(+7)-4(-7)-5(7) \\
& =0
\end{aligned}
\]
\(\displaystyle \therefore\) lines \(\displaystyle l_{1}\) and \(\displaystyle l_{2}\) are coplanar
(ii)
Any point on \(\displaystyle l_{1}:(3 \lambda-1,-2 \lambda+3,-\lambda-2)\)
Substituting in equation of \(\displaystyle l_{2}\),
\[\frac{3 \lambda-1}{-1}=\frac{-2 \lambda+3-7}{3}=\frac{-\lambda-2+7}{-2}
\]
From the first two ratios,
\[\begin{aligned}
-3(3 \lambda-1) & =-2 \lambda-4 \\
-9 \lambda+3 & =-2 \lambda-4 \\
\lambda & =1
\end{aligned}
\]
The third ratio gives the same value \(\displaystyle (-2)\), so the point is common to both lines.
\(\displaystyle \therefore\) Point of intersection is \(\displaystyle (2,1,-3)\)
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CBSE Class 12 Mathematics past-paper question from the 2022 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.