CBSE 2022 · Region 1 · Set 2 · Q14 · 4 marks
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 $\displaystyle l_{2}$.
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 \(\displaystyle (2,1,-3)\)
Three Dimensional GeometryShortest Distance between Two LinesApplycase_studymedium
More from Three Dimensional Geometry
- Find the image A^′ of the point A(1,6,3) in the line x/1=(y-1)/2=(z-2)/3. Also, find the equation of the line…2025 · asked 3×
- Find the values of λ, for which the distance of point (2,1, λ) from plane 3 x+5 y+ 4 z=11 is 2 √2 units.2022 · asked 3×
- If a line makes an angle of (π)/4 with the positive directions of both x -axis and z -axis, then the angle…2024 · asked 3×
- Assertion: If a line makes angles α, β, γ with positive direction of the coordinate axes, then sin^2 α+ sin^2…2023 · asked 3×
- If the lines (x-3)/1=(1-y)/1=(z+2)/(p) and (2-x)/3=(y+1)/5=(z+56)/(2 p) are perpendicular to each other, then…2026 · asked 3×
- Find the equation of the plane passing through points (2, 1, 0), (3, -2, -2) and (1, 1, -7). Also, obtain its…2022 · asked 3×
- Direction ratios of a vector parallel to line x-1/2=-y=(2 z+1)/6 are:2024 · asked 3×
- Assertion ( A ): The lines r= a 1+λ b 1 and r= a 2+μ b 2 are perpendicular, when b 1 · b 2=0. Reason (R): The…2023 · asked 3×
CBSE Class 12 Mathematics past-paper question from the 2022board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.