JEE Main 2022 (Online) 26th July Evening ShiftQuestion 1
The largest value of $a$, for which the perpendicular distance of the plane containing the lines $\vec r=(\hat i+\hat j)+\lambda(\hat i+a\hat j-\hat k)$ and $\vec r=(\hat i+\hat j)+\mu(-\hat i+\hat j-a\hat k)$ from the point $(2,1,4)$ is $\sqrt3$, is ________.
3D Geometry
Answer$2$
Solution
Directions $(1,a,-1)$, $(-1,1,-a)$, common point $(1,1,0)$. Normal $=(-a^2+1,\,a+1,\,a+1)$ (needs $a\ne-1$). Vector from $(1,1,0)$ to $(2,1,4)$ is $(1,0,4)$; dot with normal $=5+4a-a^2$. $\frac{(5+4a-a^2)^2}{(1-a^2)^2+2(a+1)^2}=3$ has roots $a=-4,-1,2$; $a=-1$ is rejected (directions parallel, no unique plane). Largest $a=2$.