JEE Main 2022 (Online) 28th July Evening ShiftQuestion 1
A plane $P$ is parallel to two lines whose direction ratios are $-2,1,-3$ and $-1,2,-2$ and it contains the point $(2,2,-2)$. Let $P$ intersect the co-ordinate axes at the points $A,B,C$ making the intercepts $\alpha,\beta,\gamma$. If $V$ is the volume of the tetrahedron $OABC$, where $O$ is the origin, and $p=\alpha+\beta+\gamma$, then the ordered pair $(V,p)$ is equal to :
3D Geometry
Solution
Normal $=(-2,1,-3)\times(-1,2,-2)=(4,-1,-3)$. Plane through $(2,2,-2)$: $4(2)-2-3(-2)=12$, so $4x-y-3z=12$. Intercepts: $\alpha=3,\ \beta=-12,\ \gamma=-4$. $V=\frac16|\alpha\beta\gamma|=\frac{144}{6}=24$; $p=3-12-4=-13$. $(V,p)=(24,-13)$. Option (B).