JEE Main 2022 (Online) 28th July Evening ShiftQuestion 1
Let the lines $\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}$ and $\frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}$ be coplanar and $P$ be the plane containing these two lines. Then which of the following points does NOT lie on $P$?
3D Geometry
Solution
Coplanarity: vector between points $(-27,-20,-31)$ with directions $(\lambda,1,2)$, $(-2,3,\lambda)$: determinant $=0\Rightarrow\lambda=3$. Normal $=(3,1,2)\times(-2,3,3)=(-3,-13,11)$; plane through $(1,2,3)$: $3x+13y-11z+4=0$. Test: $(0,-2,-2)$: $-26+22+4=0$ yes; $(-5,0,-1)$: $-15+11+4=0$ yes; $(3,-1,0)$: $9-13+4=0$ yes; $(0,4,5)$: $52-55+4=1\ne0$ no. Option (D).