JEE Main 2023 (Online) 6th April Evening ShiftQuestion 1
If the lines $\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}$ and $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}$ intersect, then the magnitude of the minimum value of $8\alpha\beta$ is $\_\_\_\_\_\_$ .
3D Geometry
Answer$18$
Solution
Line 1: $(1+2s,\,2+3s,\,3+\alpha s)$. Line 2: $(4+5m,\,1+2m,\,\beta m)$. $1+2s=4+5m$ and $2+3s=1+2m$ give $s=m=-1$. z-coordinates: $3-\alpha=-\beta\Rightarrow\beta=\alpha-3$. $8\alpha\beta=8\alpha(\alpha-3)=8(\alpha-\tfrac32)^2-18$, with minimum $-18$. Magnitude $=18$.