JEE Main 2022 (Online) 27th July Morning ShiftQuestion 1
If the plane $P$ passes through the intersection of two mutually perpendicular planes $2x+ky-5z=1$ and $3kx-ky+z=5$, $k<3$ and intercepts a unit length on positive $x$-axis, then the intercept made by the plane $P$ on the $y$-axis is
3D Geometry
Solution
Perpendicular normals: $6k-k^2-5=0\Rightarrow k=1,5$; $k<3\Rightarrow k=1$. Planes $2x+y-5z-1=0$ and $3x-y+z-5=0$. Family: $(2+3\lambda)x+(1-\lambda)y+(\lambda-5)z=1+5\lambda$. $x$-intercept $=\frac{1+5\lambda}{2+3\lambda}=1\Rightarrow\lambda=\frac12$. $y$-intercept $=\frac{1+5\lambda}{1-\lambda}=\frac{7/2}{1/2}=7$. Option (D).