JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Three Dimensional Geometry

For $L$ the direction is $(-2,0,1)$ through $(5,\lambda ,-\lambda )$, and for $L_1$ the direction is $(1,1,-1)$ through $(-1,1,4)$; their cross product $(-1,-1,-2)$ has magnitude $\sqrt{6}$. The shortest distance is $\frac{|\lambda +3|}{\sqrt{6}}$, and equating it to $2\sqrt{6}$ gives $|\lambda +3|=12$, so $\lambda =9$ since $\lambda \ge 0$. A general point of $L$ is then $(5-2k,9,k-9)$, for which $\alpha +2\gamma =(5-2k)+2(k-9)=-13$, a constant. This can never equal $24$, so $\alpha +2\gamma =24$ is the impossible relation.