JEE Main 2021 (Online) 22nd July Evening ShiftQuestion 1
Let $L$ be the line of intersection of planes $\vec{r}\cdot\left(\hat{i}-\hat{j}+2\hat{k}\right)=2$ and $\vec{r}\cdot\left(2\hat{i}+\hat{j}-\hat{k}\right)=2$. If $P(\alpha,\beta,\gamma)$ is the foot of perpendicular on $L$ from the point $(1,2,0)$, then the value of $35(\alpha+\beta+\gamma)$ is equal to :
3D Geometry
Solution
Direction $(1,-1,2)\times(2,1,-1)=(-1,5,3)$, point $(\frac43,-\frac23,0)$. Projection parameter $\frac{41/3}{35}=\frac{41}{105}$, foot $=\frac1{105}(99,135,123)$. Sum $=\frac{357}{105}$, so $35(\alpha+\beta+\gamma)=119$.