JEE Main 2023 (Online) 30th January Evening ShiftQuestion 1
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
3D Geometry
Solution
Line direction $(1,1,-1)$. Normal $=(3,0,-1)\times(2,1-k,2)=(1-k,-8,3-3k)$; dot with direction $=0$ gives $2k-10=0$, $k=5$. Value $=\frac{26}{12}=\frac{13}6$.