If two lines L1 : x-1/1=y-2/-1=z-1/2; L2 : x+1/-1=y-2/2=z/1.

Question

If two lines \(L_1: \frac{x-1}{1}=\frac{y-2}{-1}=\frac{z-1}{2}; L_2: \frac{x+1}{-1}=\frac{y-2}{2}=\frac{z}{1}.\) Let the line \(L_3\) passes through the point \((\alpha, \beta, \gamma)\ \text{such that} \ L_3\) is perpendicular to \(L_1\ \text{to}\ L_2\) and \(L_3\) intersects \(L_1.\) Then \(|5 \alpha-11 \beta-8 \gamma|\) is equal to

(1) 18

(2) 25

(3) 16

(4) 20

Answer 1

Correct option is (2) 25 

Let the L₃ be

\(\frac{x-\alpha}{a}=\frac{y-\beta}{b}=\frac{z-\gamma}{c},(a \hat{i}+\hat{b}+c \hat{k})\) is parallel to

\((\hat{i}-\hat{j}+2 \hat{k}) \times(-\hat{i}+2 \hat{j}+\hat{k}) \)

(a, b, c) = (5, 3, 1) 

\(\Rightarrow \frac{x-\alpha}{5}=\frac{y-\beta}{3}=\frac{z-\gamma}{-1}\)

\(\Rightarrow \text{Let the point of intersection be}\ P.\)

\(\Rightarrow 5 \lambda+\alpha=P+1,3 \lambda+\beta=P+2,-\lambda+\gamma=2 P+1 \)

\(\Rightarrow \alpha=(P+1-5 \lambda), \beta=(-P+2-3 \lambda), \gamma=(2 P+1+\lambda) \)

\(\Rightarrow|5 \alpha-11 \beta-8 \gamma|=|-25|=25\)