MHTCET 18 Apr 2026 Shift 1 Paper

Concept:Two lines intersect when their parametric coordinates become equal for some values of the parameters.Explanation:Write the first line using parameter $\lambda$:$$x=3+\lambda,\quad y=5+2\lambda,\quad z=1-\lambda$$Write the second line using parameter $\mu$:$$x=4+2\mu,\quad y=2-\mu,\quad z=4+2\mu$$At the point of intersection, the $x$, $y$, and $z$ coordinates must be equal.Equating $x$-coordinates:$$3+\lambda=4+2\mu \Rightarrow \lambda-2\mu=1$$Equating $z$-coordinates:$$1-\lambda=4+2\mu \Rightarrow \lambda+2\mu=-3$$Adding the two equations:$$(\lambda-2\mu)+(\lambda+2\mu)=1+(-3)$$$$2\lambda=-2 \Rightarrow \lambda=-1$$Substitute $\lambda=-1$ into $\lambda+2\mu=-3$:$$-1+2\mu=-3 \Rightarrow \mu=-1$$Put $\lambda=-1$ in the first line:$$x=3+(-1)=2,\quad y=5+2(-1)=3,\quad z=1-(-1)=2$$So the point of intersection is $(2,3,2)$.This satisfies both lines, hence it is the correct point.Answer:Option A: $(2,3,2)$