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Read the passage and answer the questions.
In the beautiful town of Darjeeling in the Himalayan foothills, the city planning committee wants…
In the beautiful town of Darjeeling in the Himalayan foothills, the city planning committee wants to construct two major roads to connect the various neighbourhoods. The two roads are represented by the equations $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ and $\frac{x-2}{1} = \frac{y-4}{k} = \frac{z-6}{7}$. As an in charge of the planning committee, ensure that these roads lie on the same plane to facilitate efficient urban planning and infrastructure development.
(a)[2.0]
For what value of k, will the construction meet the requirement?
(b)[2.0]
Hence, find the equation of the plane containing these lines.
Answer
Answer (a)
Official answer keyRequirement will be met if the lines are coplanar. To be coplanar,
$\begin{vmatrix}2-(-1) & 4-(-3) & 6-(-5)\\3 & 5 & 7\\1 & k & 7\end{vmatrix} = 0$
$\begin{vmatrix}3 & 7 & 11\\3 & 5 & 7\\1 & k & 7\end{vmatrix} = 0$
$3(35 - 7k) - 7(21 - 7) + 11(3k - 5) = 0$
$105 - 21k - 147 + 49 + 33k - 55 = 0$
$12k = 48 \Rightarrow k = 4$
Answer (b)
Official answer keyThe equation of the plane is:
$\begin{vmatrix}x+1 & y+3 & z+5\\3 & 5 & 7\\1 & 4 & 7\end{vmatrix} = 0$
$(x+1)(35-28) - (y+3)(21-7) + (z+5)(12-5) = 0$
$x - 2y + z = 0$.
Final answer: $x - 2y + z = 0$
From ISC 2025 Practice Mathematics, question 112.
Check your working with the 3D geometry and vectors calculator: points, vectors, lines and planes: distances, angles, foot and image, shortest distance, intersections.