JEE Main 2023 (Online) 24th January Evening ShiftQuestion 1
If the foot of the perpendicular drawn from $(1,9,7)$ to the line passing through the point $(3,2,1)$ and parallel to the planes $x+2y+z=0$ and $3y-z=3$ is $(\alpha,\beta,\gamma)$, then $\alpha+\beta+\gamma$ is equal to
3D Geometry
Solution
Direction $(1,2,1)\times(0,3,-1)=(-5,1,3)$. Foot $(3-5t,2+t,1+3t)$: $(2-5t,-7+t,-6+3t)\cdot(-5,1,3)=35t-35=0$, $t=1$. Foot $(-2,3,4)$, sum $5$.