Let the line L pass through $(1,1,1)$ and intersect the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z}{1}$ . Then, which of the following points lies on the line L ?

• A

  $(4,22,7)$
• B

  $(5,4,3)$
• C

  $(10,-29,-50)$
• Correct answer
  D

  $(7,15,13)$

Solution

$\begin{aligned}<br/>& \text { Dr's of } \mathrm{AC} \Rightarrow 2 \lambda, 3 \lambda-2,4 \lambda \\ & \text { Dr's of } \mathrm{BC} \Rightarrow \mu+2,2 \mu+3, \mu-1 \\ & \Rightarrow \frac{\mu+2}{2 \lambda}=\frac{2 \mu+3}{3 \lambda-2}=\frac{\mu-1}{4 \lambda} \\ & \Rightarrow 2(\mu+2)=\mu-1 \Rightarrow \mu=-5 \\ & \Rightarrow \text { Dr's of } \mathrm{BC} \Rightarrow 3,7,6 \\ & \Rightarrow \text { equation of } \mathrm{L} \Rightarrow \frac{\mathrm{x}-1}{3}=\frac{\mathrm{y}-1}{7}=\frac{\mathrm{z}-1}{6}<br/>\end{aligned}$
$(7,15,13) \text { satisfies. }$

Hence, the correct answer is option: 'D'.