Question

The vector equations of the two lines $L_1$ and $L_2$ are given by
$L_1:\overrightarrow{r}=2\hat{i}+9\hat{j}+13\hat{k}+\lambda (\hat{i}+2\hat{j}+3\hat{k});L_2:\overrightarrow{r}=-3\hat{i}+7\hat{j}+p\hat{k}+\mu (-\hat{i}+2\hat{j}-3\hat{k})$then the lines $L_1$ and $L_2$ are

• A. skew lines for all $pϵR$
• B. intersecting for all $pϵR$ and the point of intersection is $(-1,3,4)$
• C. intersecting lines for $p=-2$
• D. intersecting for all real $pϵR$

Answer 1

Answer: (C)

intersecting lines for $p=-2$

$L_1=2\hat{i}+9\hat{j}+13\hat{k}+\gamma (\hat{i}+2\hat{j}+3\hat{k})$ and

$L_2=-3\hat{i}+7\hat{j}+P\hat{k}+H(\widehat{-i}+2\hat{j}+3\hat{k})$

direction ratio for first line $(1,2,3)$ and direction ratio for second line $(-1,2,3)$

Cortesianform of line $L_1\to \frac{x-2}{1}=\frac{y-9}{2}=\frac{z-13}{3}=K$

$\Rightarrow x=k+2$......(i)

$y=2k+9$.....(ii)

$z=3k+12$.....(iii)

$L_2\to \frac{x+3}{-1}=\frac{y-T}{2}=\frac{z-P}{-3}$

if they are interesting

from $L_2\to \frac{K+2+3}{-1}=\frac{2k+9-7}{2}$

$\frac{k+5}{-1}=\frac{2k+2}{2}\Rightarrow 2K+10=-2k$

$4K=12$

$K=-3$

and from $\left(i\right),\left(ii\right)$ and $\left(iii\right)$

$x=-1$

$y=3$

$z=4$

Putting in $L_2$

$\to \frac{3-7}{2}=\frac{4-P}{-3}$

$\to -2=\frac{4-P}{-3}$

$\to +6=4-P$

$\to P=-2$

here for $P=-2$ line intersecting