3D Geometry
Plane through given points and parallel to a line
nta_pyq_2023_jan
Grade 12
Question:
If a plane passes through the points $(-1, k, 0)$, $(2, k, -1)$, $(1, 1, 2)$ and is parallel to the line $\frac{x-1}{1} = \frac{2y+1}{2} = \frac{z+1}{-1}$, then the value of $\frac{k^2+1}{(k-1)(k-2)}$ is
$\frac{17}{5}$
$\frac{5}{17}$
$\frac{6}{13}$
$\frac{13}{6}$
Step-by-Step Solution
Key Concept: Find normal to plane (using $\overrightarrow{CA}\times\overrightarrow{CB}$), impose that it is perpendicular to direction of given line.
Cross product gives normal $(1-k)\hat{i} - 8\hat{j} + (3-3k)\hat{k}$. Line direction $(1,1,-1)$. Dot product: $(1-k)-8-(3-3k)=0 \Rightarrow 2k=10 \Rightarrow k=5$. $\frac{26}{12} = \frac{13}{6}$. Answer: (4)
Correct Answer: $\frac{13}{6}$