JEE Main 2023 (Online) 30th January Morning ShiftQuestion 1
If $\lambda_1<\lambda_2$ are two values of $\lambda$ such that the angle between the planes $P_1:\vec r\cdot\left(3\hat i-5\hat j+\hat k\right)=7$ and $P_2:\vec r\cdot\left(\lambda\hat i+\hat j-3\hat k\right)=9$ is $\sin^{-1}\left(\frac{2\sqrt6}{5}\right)$, then the square of the length of perpendicular from the point $\left(38\lambda_1,10\lambda_2,2\right)$ to the plane $P_1$ is ______.
3D Geometry
Answer$315$
Solution
$\cos\theta=\frac15$: $25(3\lambda-8)^2=35(\lambda^2+10)$ gives $\lambda=\frac{25}{19},5$. Point $(50,50,2)$; distance to $3x-5y+z=7$ is $\frac{|150-250+2-7|}{\sqrt{35}}$, square $=\frac{105^2}{35}=315$.