JEE Main 2022 (Online) 28th June Morning ShiftQuestion 1
Let the plane $P:\vec{r}\cdot\vec{a}=d$ contain the line of intersection of two planes $\vec{r}\cdot\left(\hat{i}+3\hat{j}-\hat{k}\right)=6$ and $\vec{r}\cdot\left(-6\hat{i}+5\hat{j}-\hat{k}\right)=7$. If the plane $P$ passes through the point $\left(2,3,\frac{1}{2}\right)$, then the value of $\frac{|13\vec{a}|^{2}}{d^{2}}$ is equal to
3D Geometry
Solution
Plane $\vec r\cdot[(1,3,-1)+k(-6,5,-1)]=6+7k$. Through $(2,3,\tfrac12)$: $(2+9-\tfrac12-6)+k(-12+15-\tfrac12-7)=0\Rightarrow \tfrac92-\tfrac92k=0$, $k=1$. $\vec a=(-5,8,-2)$, $d=13$. $|\vec a|^2=93$, so $\frac{|13\vec a|^2}{d^2}=\frac{169\cdot93}{169}=93$. Option (B).