Let L₁ and L₂ denoted the lines {r}={i}+λ(-{i}+2{j}+2{k}),λ R and {r}=(2{i}-{j}+2{k}), R respectively,If L₃ is a line which is perpendicular to both L₁ and L₂ and cuts both of them, then which of the following options describe(s) L₃? [JEE Adv 2019 P1]

$\vec{r}=\;\frac{2}{9}(2\hat{i}-\hat{j}+2\hat{k})+t(2\hat{i}+2\hat{j}-\hat{k}),t\in R$
$\vec{r}=t(2\hat{i}+2\hat{j}-\hat{k}),t\in R$
$\vec{r}=\;\frac{2}{9}(4\hat{i}+\hat{j}+\hat{k})+t(2\hat{i}+2\hat{j}-\hat{k}),t\in R$
$\vec{r}=\;\frac{1}{3}(2\hat{i}+\hat{k})+t(2\hat{i}+2\hat{j}-\hat{k}),t\in R$

Solution:

L_1:\bar{r}=\bar{i}+(-i+2j+2k)

..........(1)

\lambda\in R

L_2=\bar{r}=\mu(2i-j+2k)

.......(2)

\mu\in R

given lines

Drs if required line

L_3

(-1,2,2)\times(2,-1,2)

i.e

2,2,-1

Let P on L be

(1-\lambda,2l,2\lambda)

Q on be

(2\mu,\mu,2\mu)

\overline{PQ}\parallel(2,2,-1)

\therefore\;\;\frac{1-\lambda-2\mu}{2}=\;\;\frac{2\lambda+\mu}{2}=\;\;\frac{2\lambda-2\mu}{-1}

Solving

3\lambda+3\mu=1

......(1)

\lambda=\frac{1}{9}

6\lambda-3\mu=0

............(2)

\mu=\frac{2}{9}

\therefore P=\left(\;\frac{8}{9},\;\frac{2}{9},\;\frac{2}{9}\right), Q=\left(\;\frac{4}{9},-\;\frac{2}{9},\;\frac{4}{9}\right)

From choices 1, 3, 4 are correct.

Go to Question:

Prev Question

Next Question